Structure and Measure of Linear Point Sets: Preface. Chapter One

V. L. Nekrasov.

Structure and Measure of Linear Point Sets: Preface. Chapter One

V. L. Nekrasov.

Submitted 1907 | SovietRxiv: ru-190701.07351 | Translated from Russian

Pre-reform orthography. The original is printed in the Russian spelling used before the 1918 reform (ѣ, і, ъ).

Full Text

Proceedings

of the Tomsk Technological Institute

of Emperor Nicholas II.

Vol. 5. 1907. No. 2.

V. L. Nekrasov.

Structure and Measure of Linear Point Domains.

Preface. Chapter One. I–VIII, 1–102.

The theory of point sets, created by G. Cantor, by the beginning of the twentieth century already constituted an influential branch of mathematics, a branch that with each passing day was widening the circle of its applications. But all the data of this theory were scattered through various journals, and there was no treatise in which all the results of the theory would have been collected and classified¹. This task was partly carried out by E. Borel in 1898 in his Leçons sur la théorie des fonctions, while pursuing his own special aims; and only in 1900 did Schoenflies, in the report presented to the German Mathematical Society, and a year earlier—in the Encyklopädie der Mathematischen Wissenschaften—give a systematic exposition of the whole doctrine as it stood by 1900.

Pursuing the aims of systematization, Schoenflies was almost not interested in the historical perspective of the development of the whole doctrine; therefore, if one begins acquaintance with the theory of sets only from Schoenflies’s report, it is difficult to imagine how this doctrine developed, and to whom the appearance of one or another new concept is due.

The first and third chapters of the present work aim to supplement Schoenflies’s work in this respect, since they deal with the structure and measure of point sets, but do not concern the theory of transfinite numbers or applications of the theory of sets to the theory of functions and geometry². The historical survey contained in these chapters will, I hope, give a sufficiently complete picture of the development of the doctrine from its very origin down to the most recent time.

Following the growth of the new theory, I could be very concise so long as the discussion concerned works of an earlier period, partly because their relation to the theory of sets was comparatively remote—if one does not count the works of G. Cantor—and partly because everything more important in these works had been set forth with exhaustive completeness in Schoenflies’s report; but, as I passed to more recent works, especially those that appeared after this report, I tried to extract from them everything essential,

—so that, in addition to a historical survey, my work might to some extent be a continuation of the corresponding chapters of Schoenflies’s report; at the same time I had to make certain changes in the proofs of the theorems, to add certain theorems, and to propose some changes in the formulations and definitions. In this way the first and third chapters, according to my design, were to serve partly as a supplement—from another point of view—to Schoenflies’s work, and partly as its continuation.

In the second chapter, devoted to the structure of a linear domain, having gathered all the results established up to now, I pass to the study of three types of arrangement $\omega$, ${}^{*}\omega$, $\breve{\omega}$, and show in what manner a finite or transfinite combination of these types gives domains of ever-increasing complexity, possessing a structure completely characterized by their type of arrangement. At the end of the chapter I show that for every given closed domain a quite definite type can be indicated; as for non-closed domains, for some of them such a type can also be determined, but it has not yet been possible to prove that this type exists for every non-closed domain. By using types of arrangement, we can characterize a continuous function of a real variable by a certain symbol indicating the arrangement of its points of discontinuity.

Concerning the content of the fourth chapter, I must note that, owing to a number of external circumstances, the printing of the present work, begun in the autumn of 1904, extended over two and a half years. It is, of course, understandable that during this time new investigations appeared in print which could not be ignored and which could not fail to affect the content of my work. According to its original plan, this work was to consist of three chapters, of which the second and third were devoted to the structure and measure of a domain, and the first to the history of the development of these theories. In view of the circumstances mentioned above, it became necessary to add to the first chapter a new third chapter, devoted to the latest work in the corresponding fields of science; as for the former third chapter, now turned into the fourth, it underwent, in comparison with the original intention, an almost complete revision. To preserve its beginning in the form in which it had been planned earlier, I, in view of W. H. Young’s exhaustive work on the theory of the measure of non-closed domains, abandoned my attempts in this direction. Thus I had either to discard this last chapter altogether from my work, thereby disrupting the plan that had originally been...

—was compiled, and at the proposal of which the first chapter was set forth and has already been printed; or else—to keep this chapter and follow Young in the theory of measure. I settled on the latter course and decided to put everything concerning the measurement of sets into systematic form, so as to give the corresponding material the form of a sufficiently complete “theory of measure,” which title I have also given to this chapter. In it, therefore, we find only a unification of the results obtained up to now, while the final solution of the problem of measure is connected with the investigation of the character of a certain non-closed set, which I have called elementary; this elementary set differs somewhat from the non-closed set of which Young speaks, in the sense of indicating several properties which it must possess; but whether it possesses the theorem of internal composition, which is the decisive point in the theory of the measure of a set, still remains an open question.

Then, according to my original plan, I intended to devote my work not only to linear sets, but also to sets of two dimensions; traces of this intention occur in the first chapter; but it later turned out that, even without the inclusion of the latter sets, the work assumed rather considerable dimensions; moreover, in recent times a number of investigations have appeared concerning the theory of two-dimensional sets and straying too far from the circle of ideas to which the present work is devoted. I have therefore considered it best to restrict its plan, leaving two-dimensional sets outside its boundaries, even though I had occupied myself with the question of these sets even before the plan of the work in its present form had taken shape.

I shall also note that I have only just read of the appearance in London of the work by W. H. Young and G. C. Young, “Theory of sets of points,” which is likely to give a systematic exposition of the whole theory of sets; but I do not consider it possible to delay any longer the publication of the present work, the chief part of which is the second chapter; “the theory of sets—according to Schoenflies’s expression¹—is a kind of molecular theory of mathematical magnitudes,” and this molecular investigation of a point set and its reconstitution from the elements $\omega$, $\omega^*$, $\hat{\omega}$ is precisely the aim of the second chapter. And since² “the theory of sets—in the future everything, but its influence grows with each day,” as is proved by all the increa—

the growing number of articles devoted to it, I permit myself to hope that the study of types of arrangement, even in its present form, will not be useless for elucidating the structure of sets.

In appending at the end of the book the literature on the theory of sets, I have indicated all the works known to me—even, for the sake of completeness, those which have no direct relation to the theory of the structure and measure of a linear set, but concern only the theory of transfinite numbers or of two-dimensional sets; these latter works are marked by me with an asterisk, and those with which I was unable to acquaint myself are marked with a small circle.

Tomsk,
9 January 1907.

Corrections

Page	Line	Printed	Should read
25	6 from top	$m,$	$m_n,$
25	10 from top	$(m_1 + m_2 + \cdots + m_n)$	$\dfrac{1}{n}(m_1 + m_2 + \cdots + m_n)$
25	14–16 from top	$m_n$	$m_n,$
55	6 from bottom	$i;$ Rona	$P_i;$ it
78	3 from bottom	$\displaystyle \sum i$	$\displaystyle \sum l'_{2i}$
92	12 from bottom	middle	internal for
92	7 from bottom	$P^{(0)}$ will be	$P^{(0)}$, not belonging to $P$, will be
99	13 from top	Peano, Jordan and Lebesgue; two penultimate	Peano and Jordan; two last
100	15 from bottom	III	IV
128	9 from bottom	88. Suppose	Suppose
144	4 from bottom	$44^\circ$	$95^\circ$
208	15 from top	$y_1'$-th	$y_1'$-of

CHAPTER I.

Historical Sketch.

1. The theory of point domains is one of the youngest branches of pure mathematics; its ideas assumed a somewhat definite form only some twenty years ago.

Slowly, but surely, this theory won for itself the right to exist, and now there can no longer be any doubt as to its future role in the justification of the most delicate concepts of mathematical analysis.

Since the time of Leibnitz and Newton, who created the analysis of infinitesimals, of the two elements on which this analysis was built—the function and the argument—only the function attracted attention, whereas with regard to the argument everyone seemed tacitly to agree that there was nothing there to study.

But little by little it became clear that the matter of the argument, too, was by no means so simple as it had at first seemed; therefore due attention had to be paid to it as well. A theory of the argument arose in the form of the theory of point domains.

The concept of a domain (Menge) first appeared in Bolzano (1847); he defines¹ a domain as an aggregate (Inbegriff) whose arrangement of parts is indifferent, i.e. with respect to which nothing essential for us changes if only the arrangement changes. A domain of units of a certain kind Bolzano calls a multiplicity (Vielheit)²; he distinguishes³ finite or countable (zählbare) domains and infinite domains, and speaks⁴ of a domain of all whole numbers.

Then⁵ Bolzano says that “not all infinite domains, with respect to their multiplicity, may be considered equal among

itself”; but in this respect he did not yet arrive at establishing the concept of equivalence in Cantor’s sense, although the concept of a one-to-one correspondence between different domains is expressed by him quite precisely¹: “Two infinite domains may stand to one another in such a relation that it is possible—on the one hand—to connect every object belonging to one domain with an object of the other in such a way that not one object of these domains remains without a mate and enters into two or several different pairs; while at the same time—on the other hand—it is possible that one of these domains contains the other as a part, so that the manifolds which they represent, if we regard the objects as units, have the most diverse relations to one another.” This very relation was taken as the starting point for the investigation of infinite domains by G. Cantor² and, after him, by Dedekind³.

Next one must note⁴ the “paradoxes” connected with the concept of countability; Bolzano further recognizes⁵ that the concept of the “magnitude of an extension,” i.e. the concept of the measure of a continuous domain, requires immediate clarification; at the same time he draws attention⁶ to the significance of the boundary of domains, and to the influence⁷ on the measure of a domain of the removal of a countable sequence of points, whereby limit points first appear in science, playing so large a role in the theory of domains.

Thus it must be acknowledged that the originator of the modern theory of domains was Bolzano, but that G. Cantor developed it and placed it upon a strictly scientific foundation.

2. The principal questions that became the task of the theory of domains are the questions of the measure and the structure of a domain.

The concept of the measure of a domain arose in the investigation of the possibility of integrating a continuous function; therefore this concept appears, in one form or another, wherever a definite integral is at issue.

The first who, in 1854, directed mathematical thought toward the “measurement” of a domain must be named Riemann. In establishing the well-known condition for integrability, Riemann divides the interval $(a,b)$, in which the integration is carried out, into parts $\delta_i$ and considers⁸

the total length (Gesammtgrösse) of those intervals $\delta_i$ where the oscillation of the function exceeds some small number $\sigma$; here there is as yet no notion of measure in the strict sense, but Riemann takes another step toward establishing it: he applies his criterion to the investigation of the function

\[ f(x)=\sum_{1}^{\infty}\frac{(n x)}{n^2}, \]

which has discontinuities for all values of the variable equal to

\[ \frac{p}{2n}, \]

where $p$ and $n$ are relatively prime numbers. Here Riemann does not yet include all discontinuities in certain intervals; pursuing his aim, he does this only with respect to those points $x=\frac{p}{2n}$ at which the discontinuities are $> \sigma$, and says that the total length of such intervals can be arbitrarily small.

3. The next person to take up, in 1870, the idea thrown out by Riemann was Hankel, who formulated it more fully and precisely, establishing¹ the notion of everywhere dense domains (überall dicht—in Cantor’s terminology); the points of these he called points filling a segment (Punkte, die Strecke erfüllen), and nowhere dense (in Cantor’s terminology—nirgends dicht) points are those which are scattered on a segment (zerstreut liegen). Hankel tries to prove a theorem which may be expressed as follows: “If a domain is nowhere dense, then the total length $s$ of the intervals containing it can be arbitrarily small, and conversely.”

This theorem, as was later proved, is not correct. For us, however, it is of interest only to see by what path Hankel arrives at this total length. He says: if a domain consists of a finite number of points, $s$ is composed of intervals which lie around each of them; since each of these intervals can be made arbitrarily small, $s$ too will be arbitrarily small. If the number of points is infinite, then between them—by the definition of a nowhere dense domain—there still lie free intervals; we then divide the whole segment into intervals so that—first—each of them contains one of the points of the domain, and—second—so that taken together they fill the whole interval $(a,b)$. If then each of the intervals is

reduce it to its $\frac{1}{n}$-th part; if the first condition is satisfied, then the remaining $\frac{n-1}{n}$ part of $(a,b)$ will be free of points of the set. Thus $s$ can be made arbitrarily small.

The central point of the preceding proof consists—first of all—in the assumption that, to use modern language, the number of points of the set is countable¹, and—secondly—that one can construct intervals around each point of the set in such a way that they do not overlap one another; the latter can be carried out to some extent for a nowhere dense countable set: namely, for all its isolated points the construction of these intervals is possible; as regards limiting points, the intervals corresponding to such points cannot fail to contain other intervals. Nevertheless, despite this fact, for these sets $s$ can be arbitrarily small, as can be proved in another way. As for the first, unconscious, assumption of Hankel, it is completely arbitrary and false; it is known that there exist uncountable nowhere dense sets—namely, perfect ones—for which Hankel’s construction of intervals becomes impossible.

It is of interest to note that Hankel’s idea concerning the construction of intervals around each of the points of a given set subsequently found its development in the works of Borel and there yielded very interesting results.

4. Then, from 1872 onward, G. Cantor entered upon the path of studying point sets, when he², as the basis for his further investigations, put forward the proposition that “to each numerical value there corresponds a unique point of the line, and conversely”; in this way the one-dimensional continuum and the geometry of the line were brought into close connection. In this same article Cantor establishes the notion of a limiting point and of derived sets, classifying sets by kinds according to the existence of a certain sequence of derived sets.

In the same year³ Cantor for the first time draws attention to the countability of sets, establishing a one-to-one correspondence between sets of algebraic and of whole numbers and proving the uncountability of all real numbers.

Until the present moment the study of sets had borne an accidental character; but after Cantor gave the definition of a limiting point and of the derivative of a set, the theory of sets had the beginnings of becoming an independent theory with its own special content and its own methods of investigation.

5. Further, in 1875¹, in Smith’s work the question of the measure of a set receives a new clarification; Smith gives several examples of rarely distributed sets and computes the total sum of the intervals occupied by them.

A. Construct on the interval $(0, 1)$ rarely distributed sets

\[ \mathrm{P}_1=\left\{\frac{1}{n'}\right\},\quad \mathrm{P}_2=\left\{\frac{1}{n'}+\frac{1}{n''}\right\},\ldots,\quad \mathrm{P}_m=\left\{\frac{1}{n'}+\frac{1}{n''}+\cdots+\frac{1}{n^{(m)}}\right\}, \]

where, independently of one another,

\[ n^{(i)}=1,2,3,4,\ldots,n,\ldots, \]

and where the points of each $\mathrm{P}_{i-1}$ serve as limiting points for $\mathrm{P}_i$. Smith includes the points $\mathrm{P}_m$ in intervals in the following manner: he first takes an interval $\delta_0$ to the right of the point $0$. In the interval $(\delta_0,1)$ there is a finite number $n_1$ of points of the set $\mathrm{P}_1$; to the right of each of these points $\mathrm{P}_1$ we construct intervals $\delta_1$, the sum of which will be $n_1\delta_1$; in each of the intervals remaining after this, and consequently in all of them taken together, there lies a finite number $n_2$ of points $\mathrm{P}_2$; to the right of each of them we construct intervals $\delta_2$; and so on. In this way all the points $\mathrm{P}_m$ will lie inside or on the boundaries of a finite number

\[ \mathrm{N}=1+n_1+n_2+\cdots+n_m \]

of intervals, whose total sum will be

\[ \delta_0+n_1\delta_1+n_2\delta_2+n_3\delta_3+\cdots+n_m\delta_m=\delta; \]

if, for a given $\varepsilon$, one takes

\[ \delta_0<\frac{\varepsilon}{3(m+1)},\qquad \delta_i<\frac{\varepsilon}{3(m+1)n_i}, \]

then it will be

\[ \delta<\frac{\varepsilon}{3}. \]

Let $\bar{\delta}$ denote the smallest of the intervals $\delta_i$; construct to the right and to the left of each of them still further intervals $\delta$; then all the points $\mathrm{P}_m$

will lie inside new intervals, whose total sum will be

\[ \delta+2N\cdot\bar{\delta}<3\delta<\varepsilon. \]

B. The segment $(0,1)$ is divided into $m$ parts; the first $m-1$ of them are again divided into $m$ parts; the same process is applied to each of them, and so on, so that after each division $\frac{1}{m}$ of the preceding interval remains excluded; the points of division give a domain $P$. For it, the sum of the excluded intervals, which—consequently—will not contain within themselves points of $P$, will be

\[ \frac{1}{m},\quad \frac{1}{m}+\frac{m-1}{m^{2}},\quad \frac{1}{m}+\frac{m-1}{m^{2}}+\frac{(m-1)^{2}}{m^{3}},\ldots, \]

or

\[ 1-\left(1-\frac{1}{m}\right),\quad 1-\left(1-\frac{1}{m}\right)^{2},\ldots,\quad 1-\left(1-\frac{1}{m}\right)^{n}; \]

as $n$ increases without bound,

\[ \lim\left\{1-\left(1-\frac{1}{m}\right)^{n}\right\}=1. \]

C. The segment $(0,1)$ is divided into $m$ parts, each of the first $m-1$ is divided into $m^{2}$, each of the $(m-1)(m^{2}-1)$ into $m^{3}$, and so on. Then the sum of the non-free intervals after the first, second, third, etc. divisions will be

\[ \frac{m-1}{m},\quad \frac{m-1}{m}\frac{m^{2}-1}{m^{2}},\ldots,\quad \frac{m-1}{m}\frac{m^{2}-1}{m^{2}}\ldots\frac{m^{n}-1}{m^{n}} \]

or

\[ 1-\frac{1}{m},\quad \left(1-\frac{1}{m}\right)\left(1-\frac{1}{m^{2}}\right),\ldots,\quad \left(1-\frac{1}{m}\right)\left(1-\frac{1}{m^{2}}\right)\ldots\left(1-\frac{1}{m^{n}}\right); \]

therefore the sum of the free intervals will be

\[ 1-\left(1-\frac{1}{m}\right),\quad 1-\left(1-\frac{1}{m}\right)\left(1-\frac{1}{m^{2}}\right),\ldots,\quad 1-\left(1-\frac{1}{m}\right)\left(1-\frac{1}{m^{2}}\right)\ldots\left(1-\frac{1}{m^{n}}\right). \]

As $n$ increases without bound, the limit of the sum $s_n$ of these free intervals,

\[ \lim s_n=1-\lim\prod_{1}^{n}\left(1-\frac{1}{m^{n}}\right)=1-\mathrm{E}\left(\frac{1}{m}\right), \]

turns out to be different from $1$; consequently, the sum of the non-free intervals, equal to $\mathrm{E}\left(\frac{1}{m}\right)$, for this sparsely scattered domain will not be equal to zero.

This last example was intended to refute Hankel’s assertion concerning rarely scattered domains.

Smith’s work is of interest in two respects: first—Smith gives the first example, for Cantor’s established notions of a limiting point and a derivative, apparently without knowing of Cantor’s works; and, secondly, he constructs domains B and C and computes a measure for them, taking as a basis intervals free of points of the domain.

6. Ascoli¹ constructs a system of frequently scattered domains

\[ \mathrm{P}_1,\ \mathrm{P}_2,\ldots,\ \mathrm{P}_{s_1},\ \mathrm{P}_{s_1+1},\ \mathrm{P}_{s_1+2},\ldots,\ \mathrm{P}_{s_2},\ldots, \]

the points of which satisfy the condition

\[ \overline{x_i x_{i+1}} < \eta_n \quad \text{for } \mathrm{P}_{s_{n-1}+1},\ \mathrm{P}_{s_{n-1}+2},\ldots,\ \mathrm{P}_{s_n} \]

for a series of infinitely decreasing positive quantities

\[ \eta_1,\ \eta_2,\ \eta_3,\ldots,\ \eta_n,\ldots. \]

In 1875 du Bois-Reymond discusses² frequently and rarely scattered domains P, condensation points (Verdichtungspunkt) of various orders, domains for which there exist $\mathrm{P}'$, $\mathrm{P}''$, and so on, and gives, as an example, the roots of Kettensin’s

\[ \operatorname{Sin}\frac{1}{\operatorname{Sin}\frac{1}{\operatorname{Sin}\cdots \operatorname{Sin}\frac{1}{\operatorname{Sin}x}}}. \]

Concerning the role of the theory of domains in Dini’s large treatise (1878), Cantor himself, mentioning the derivative $\mathrm{P}'$ of a domain $\mathrm{P}$, says³ that there “we see this concept still further developed, since it served as the point of departure for a series of remarkable generalizations of known analytic theorems.”

Incidentally, Dini was the first—it seems⁴—to state and prove⁵ the theorem:

“For a domain of the first kind, in every part of the fundamental interval there are intervals free of points of the domain; and this part may—

can be divided into intervals in such a way that the sum of the non-free intervals will be arbitrarily small.”

7. In view of Cantor’s having established¹ (1877) the concept of power (Mächtigkeit, puissance), and the possibility of putting continuous domains of $n$-dimensional space in one-to-one correspondence with a segment of a straight line, it became above all necessary to study and classify domains on the straight line, the so-called linear domains; this is what Cantor does in a series of articles in Mathematische Annalen. In 1879 Cantor introduces² domains of the first and second kind, introduces the term domain dense in an interval; he says that for such a domain $P'$ is identical with the interval, so that often dense domains will all be domains of the second kind. Here he also points out that domains of the first kind will necessarily be countable, as are also certain domains of the second kind.

In 1880 du Bois-Reymond, mentioning G. Cantor, asserts³ that he had written to Cantor about the existence of points of condensation of infinite order several years before 1880. “As for the points of condensation of incessantly decreasing segments, the order of these points being finite or infinitely large, du Bois-Reymond proposed returning later to the choice of the term pantachisch, instead of the überall dicht later introduced by Cantor.” This indeed took place in 1882 in Allgemeine Functionentheorie.

In 1880 Pincherle, in his treatise on the theory of functions, gives⁴ certain information from the theory of domains. He calls a domain of one or many dimensions a varietà, and the elements of the domain—points (posto o punto); he also speaks of a neighborhood of a point.

If for the points of a domain a neighborhood can be constructed that is free of points of the domain, then the domain forms a serie discreta.

Then Pincherle distinguishes interior points from contour points, but only for a continuum; he says that a continuous domain may have a contour but have no interior points, as for example the domain of points determined by the condition

\[ x^{2}+y^{2}+z^{2}=1. \]

Finally, following Weierstrass, he proves the existence of limiting points for a domain consisting of an infinite number of points.

8. Examples relating to the question of the measure of a domain we find¹ in a work of Volterra (1881). All with the same aim of refuting Harnack’s assertion, Volterra constructs a sparsely scattered domain as follows:

Taking on the segment $(0,1)$, at the right-hand end, an interval $(\alpha_1,1)$ equal to

\[ \frac{1}{2^2}, \]

he places between $0$ and $\alpha_1$ a countable series of points

\[ \cdots,\ \alpha_n,\ \ldots,\ \alpha_3,\ \alpha_2,\ \alpha_1 \]

in such a way that they have $0$ as a limiting point,

\[ \lim_{n\ldots\infty}\alpha_n=0. \]

The interval $(\alpha_1,1)$ remains free; subsequently no further points of the domain are distributed on it.

On each of the remaining segments $(\alpha_{n+1},\alpha_n)$ we again place a countable series of points

\[ \cdots,\ \alpha_{nn'},\ \ldots,\ \alpha_{n3},\ \alpha_{n2},\ \alpha_{n1}, \]

for which—first—$\alpha_{n+1}$ serves as limiting point,

\[ \lim_{n'\ldots\infty}\alpha_{nn'}=\alpha_{n+1}, \]

and—second—the interval $(\alpha_{n1},\alpha_n)$ constitutes the

\[ \frac{1}{2^4} \]

part of the interval $(\alpha_{n+1},\alpha_n)$.

We leave this last interval free; in each of the remaining $(\alpha_{nn'+1},\alpha_{nn'})$ we again place, in a countable series,

\[ \cdots,\ \alpha_{nn'n''},\ \ldots,\ \alpha_{nn'3},\ \alpha_{nn'2},\ \alpha_{nn'1}, \]

for which

\[ \lim \alpha_{nn'n''}=\alpha_{nn'+1},\qquad (\alpha_{nn'1},\alpha_{nn'})=\frac{1}{2^6}(\alpha_{nn'+1},\alpha_{nn'});\quad \text{etc.} \]

It is clear that the domain $\{a\}$ obtained in this process will be sparsely scattered. For this domain the sums of the lengths of the free and non-free segments will be, for the first, second, ..., $m$-th division, as follows:

free intervals	non-free
1) $\dfrac{1}{2^2},$	$1-\dfrac{1}{2^2},$
2) $\left(1-\dfrac{1}{2^2}\right)\cdot\dfrac{1}{2^4},$	$1-\dfrac{1}{2^2}-\dfrac{1}{2^4}\left(1-\dfrac{1}{2^2}\right)=\left(1-\dfrac{1}{2^2}\right)\left(1-\dfrac{1}{2^4}\right),$
3) $\left(1-\dfrac{1}{2^2}\right)\left(1-\dfrac{1}{2^4}\right)\cdot\dfrac{1}{2^6},$	$\left(1-\dfrac{1}{2^2}\right)\left(1-\dfrac{1}{2^4}\right)\left(1-\dfrac{1}{2^6}\right),$
$\cdots$	$\cdots$
$m)$ $\left(1-\dfrac{1}{2^2}\right)\left(1-\dfrac{1}{2^4}\right)\cdots\left(1-\dfrac{1}{2^{2m-2}}\right)\cdot\dfrac{1}{2^{2m}}$	$\left(1-\dfrac{1}{2^2}\right)\left(1-\dfrac{1}{2^4}\right)\left(1-\dfrac{1}{2^6}\right)\cdots\left(1-\dfrac{1}{2^{2m}}\right).$

Consequently, if the process is continued indefinitely, the sum of the non-free intervals

\[ \prod_{1}^{\infty}\left(1-\frac{1}{2^{2m}}\right)=\mathrm{E}\left(\frac{1}{4}\right) \]

will be different from 0.

9. Harnack, in his “Elemente” (1881)¹, includes every point $x$ of the given domain in a neighborhood $(x-\varepsilon, x+\varepsilon)$ and is interested in the sum of these neighborhoods; if, by reducing their number, it is possible to make this sum arbitrarily small, then such a domain is called by Harnack discrete; this is a domain sparsely scattered and empty, i.e., with measure equal to zero.

If, however, these neighborhoods cannot be decreased arbitrarily, the domain—according to Harnack—is called linear. He draws attention, in M. A., to the fact that a “discrete” domain cannot be densely scattered in any arbitrarily small interval, but not conversely, and Harnack gives an example of how a “linear” domain may be distributed so that it is sparsely scattered.

Take $n$ points $x_i$ on $(a,b)$ and make them the midpoints of intervals of length $\delta$, choosing these intervals so that

\[ n\delta < b-a. \]

On each of the intervals $\delta$ take $n'$ points and, correspondingly, intervals $\delta'$, so that

\[ n'n\delta' < n\delta < b-a \]

and so on,

\[ n^{(m)}n^{(m-1)}\ldots n''n'n\,\delta^{(m)} < n^{(m-1)}\ldots n''n'n\,\delta^{(m-1)} < \ldots < n\delta < b-a. \]

Then, depending on whether

\[ \lim n^{m} u^{(m-1)} \ldots u'' u' u\, \delta^{(m)} = 0 \quad \text{or} \quad < 0, \]

the domain of points $\{x\}$ will turn out to be “separate or linear,” and in its construction it is sparsely scattered.

10. In the papers of Veltmann (1882)$^2$ we have a further step toward establishing a more precise conception of sparsely scattered and perfect domains.

He is the first to give a general construction of a perfect domain, which must be recognized as classical, and which must serve as the foundation for the theory of domains.

In his construction Veltmann especially points out the circumstance that the infinite domains obtained in this way will be sparsely scattered; in Veltmann’s terminology this reads as follows: “not even an arbitrarily small segment is decomposed by the points of division into obviously infinitely small parts,” so that everywhere “one can take parts of finite length” free from points of the domain.

The distinctive feature of his first construction is that he constructs perfect domains on a circumference and inside a square; thus to him also belongs the first construction of a plane perfect domain.

On a circumference of radius one Veltmann takes two equal arcs, symmetrically situated and free from points of the domain; on the two remaining arcs four equal and free arcs are symmetrically placed; and so on; the sum of the free arcs may in this case be equal to $2\pi$, or less than $2\pi$, depending on the law according to which the arcs decrease. If the lengths of the free arcs of the $n$-th division are denoted by $\alpha_i^{(n)}$, and one takes

\[ \sum \alpha_i' = \frac{\pi}{2}, \qquad \sum \alpha_i'' = \frac{\pi}{2^2}, \ldots,\qquad \sum \alpha_i^{(n)} = \frac{\pi}{2^n}, \ldots, \]

then the sum of all free arcs will be $\pi$, and consequently the sum of the non-free arcs will also be equal to $\pi$.

$^2$ Zeitschrift für Mathematik, 27.

A second example, relating to the theory of plane domains, is as follows: from a square of side $l$, by means of four equal squares at the corners, we exclude a cross, inside which there are no points of the domain; from each of the four squares we again exclude crosses; and so on.

Let us establish the law of decrease of the sides $s_m$ of the squares as follows:

\[ s_m = a e^{-\frac{1}{2^m}} s_{m-1} \quad \text{for } a < \frac{1}{2}, \qquad s_0 = l; \]

then the lengths of the corresponding sides will be

\[ s_1 = a e^{-\frac{1}{2}} l,\quad s_2 = a^2 e^{-\left(\frac{1}{2}+\frac{1}{2^2}\right)} l,\ldots,\quad s_m = a^m e^{-\sum_1^m \frac{1}{2^i}} l,\ldots, \]

the area of the $m$-th square is

\[ a^{2m} e^{-2\sum_1^m \frac{1}{2^i}} l^2, \]

and the number of squares of the $m$-th division is $4^m$, so that the sum $\delta_m$ of the areas of the squares of the $m$-th division will be

\[ \delta_m = 4^m \cdot a^{2m} e^{-2\sum_1^m \frac{1}{2^i}} l^2 = (2a)^{2m} l^2 \cdot e^{-2\sum_1^m \frac{1}{2^i}}; \]

as the number of divisions increases without bound,

\[ \lim_{m..\infty} \delta_m = l^2 e^{-2}\cdot \lim (2a)^{2m} = \begin{cases} 0, & a < \dfrac{1}{2};\\[4pt] l^2 e^{-2}, & a = \dfrac{1}{2}. \end{cases} \]

If we take as the points of the domain the vertices of the crosses, then in the first case the area of the crosses, i.e. the area of the part of the square free from points of the domain, will be equal to the area of the square; in the second, the area of the free crosses will be only

\[ l^2 - l^2 e^{-2} = (1 - e^{-2})l^2, \]

so that the points of the domain cannot be included in intervals with arbitrarily small area.

In the second article Veltmann gives an example of the construction of a rarely disconnected perfect domain on the line, analogous to the construction of the preceding domains.

Concerning the last example Veltmann says: “if one calls closed such a domain which divides an interval into manifestly infinitely small segments, then the preceding domain has the peculiarity that each of its points is a limiting point¹, and nevertheless there is in it not a single closed part.”

In ordinary terminology, Veltmann’s closed domain is a set that is everywhere dense; thus Veltmann says that the domain constructed is condensed (in sich dicht)², although it is rarely dense.

In the third article, speaking of the first example, he distinguishes

1) the interior points of the free arcs,

2) their end-points, and

3) points “which never become either interior or terminal,” i.e. exterior points; such points will be the midpoints of the non-free arcs; points of the second and third categories are here limiting points.

Thus Veltmann gives to the theory of domains exact methods for constructing perfect domains, which become guiding ones in this theory. He then is the first to construct a plane, rarely dense, perfect domain.

[1.] In 1882 appeared du Bois-Reymond’s Allgemeine Functionentheorie. Subjecting here to criticism the fundamental concepts of analysis and, in particular, striving to give a theory of the argument, the author devotes much space to point domains. While conceding to Cantor priority in establishing the concepts of the extent of a domain and of connectedness, and recognizing Cantor’s merits in the study of possible groupings of the values of the argument, of their increasing condensation, in the study of the domain of all values of the argument and of its relation to more meager point domains, du Bois-Reymond says that his own works on the general theory of functions had long ago led him onto the same path, where he had to do with the distribution of points on a segment and with methods of representing irrationalities.

Du Bois-Reymond conducts his exposition from his own point of view and using his own notations; in this respect he considers himself all the more justified because he had already published something earlier, and the necessity of his basic classification of domains he in writing

communicated by Cantor more than a year before his article in M. A. 15. Therefore du Bois-Reymond regarded as his own property the general concept of a pantachia, which had arisen not from speculation, but was called forth by the needs of the theory of functions.

He points out that the values of the argument cannot be completely identified with lengths; that there is an essential difference between them, consisting in the fact that, in the former, each point appears only once; namely—if $x_0$ divides the segment $(a,b)$, then we shall have three different kinds of division, according as we assign $x_0$ to $(a,x_0)$, or to $(x_0,b)$, or to neither the one nor the other interval.

Here, in this way, we encounter for the first time a conscious distinction between open segments and closed segments, if one does not count the analogous “paradox” of Bolzano, of whose work most authors of that period—apparently—were unaware. This distinction must, in one way or another, be taken into account in establishing the concept of measure.

Passing to the distribution of points over an interval, du Bois-Reymond calls pantachian, or a pantachia, such a distribution as today bears the name of an everywhere dense domain; a nowhere dense domain he gives the name apantachian distribution; besides these and others, there are still such distributions which, falling under none of these types, do not even decompose into a finite number of pantachian or apantachian parts.

Pantachias the author further divides into the following classes¹: if a domain contains a finite number of points, increasing, however, in such a way that points appear in an arbitrarily small interval, then such a pantachia is called unbounded; these are those domains which subsequently received from Baire the name of domains of the first category. All non-unbounded pantachias du Bois-Reymond calls infinite pantachias; these will be—domains of the second category.

The continuum is a complete pantachia and belongs—obviously—to the infinite pantachias; the removal from a complete pantachia of an arbitrary number of any unbounded pantachias whatever cannot transform it into an unbounded pantachia.

To the apantachian systems belong, first of all, isolated ones, or domains of isolated points; these are systems of a finite number of points, sufficiently condensed, but not capable of being condensed arbitrarily, as in unbounded pantachias; then—systems with poin-

clusters of various orders, i.e. domains of the first kind and of the $n$-th species, or—in the terminology of du Bois-Reymond—up to points of the $n$-th order of the system’s separation.

Concerning them the author says that one can exclude the points of condensation by arbitrarily small intervals, in such a way that the sum of these intervals will be arbitrarily small. The points of condensation of du Bois-Reymond are nothing other than Cantor’s limiting points, and do not coincide with Lindelöf’s points of condensation¹. Here du Bois-Reymond makes the important observation that points of condensation of points will be counted together with points of condensation of intervals. Further, as an example of an apantachic system, du Bois-Reymond constructs a perfect domain, evidently without knowing of Veltmann’s analogous construction; at least he does not mention it.

Du Bois-Reymond attaches great importance to domains of this type, pointing out, among other things, that they have not a single isolated point.

Besides these systems, analysis also gives point-distributions which du Bois-Reymond calls integrable; these are, in modern terminology, field (unausgedehnte) systems.

One might have thought, says du Bois-Reymond, that the integrable domains coincide with the pantachic ones, but the contrary turns out to be the case: apantachic systems form the more extensive class.

Thus to du Bois-Reymond belongs the credit for introducing into science the notions of pantachy and apantachy and of distinguishing between unbounded and infinite pantachies; as for du Bois-Reymond’s terminology, it did not take root in science and was replaced by Cantor’s more convenient designations; the classification of pantachies into infinite and unbounded was destined, after a twenty-year interval, to reappear in a memoir of Baire².

12. In 1882 Cantor speaks³ of point-domains of $n$ dimensions, proves a theorem to the effect that in $n'$-dimensional space every domain consisting of or adjoining one another by intervals of $n$ dimensions will be countable, however small these intervals may be; finally he points out the fact that the removal from $n'$-dimensional space, for $n>2$, of a countable series of even very often scattered points leaves the space continuous and connected.

Further⁴ Cantor defines a connected domain $Q$ by the equality

\[ \mathrm{D}(Q,Q') = 0. \]

whereas in general

\[ P = Q + D(P, P'). \]

He then proves that the domain $P$ will be countable in the following cases: if a) it is isolated, b) $P'$ is countable, c) $P$ is of the first genus and of the $n$-th species, and d) $P^{(\omega)}$, or in general $P^{(\alpha)}$, is countable.

In this article Cantor first begins to speak of the measure of a domain. He says that du Bois-Reymond and Harnack made use of systems of points on the line which can be enclosed in a finite number of intervals with arbitrarily small total length. For this to be possible, the domain of points must not be frequently scattered in any part of the interval; but this condition alone is not sufficient.

Cantor proves that this property is possessed first of all by those domains for which $P'$ is countable. In proving this theorem, Cantor deals with the countable sequence of points

\[ \tag{1} x_1,\ x_2,\ x_3,\ \ldots,\ x_n,\ \ldots \]

and says that the points of the interval $(a,b)$ not entering into (1) can be included in an infinite sequence of intervals

\[ (c_1,d_1),\ (c_2,d_2),\ \ldots,\ (c_n,d_n),\ \ldots, \]

maximal ones—adjoining one another; Cantor then proves that the sum of these intervals has, as its limit, the length $b-a$, whence the assertion of the theorem follows.

This proof is very important, because here, as also with Veltmann, the idea was for the first time directed not to intervals including the points of the domain, but, on the contrary, to intervals free of these points.

It is evident that the ideas of Cantor and Veltmann developed, in this respect, quite independently of one another, and that both illuminated one and the same question from different points of view and thus supplemented one another: Cantor, for a given domain, constructs a sequence of free intervals, whereas Veltmann, from a given sequence of free intervals, determines a domain; Cantor is dealing here with a countable domain, whereas Veltmann is dealing with perfect domains. Thus to Veltmann and Cantor belongs the merit of having established firm foundations for the study of the structure and measure of a domain.

It is interesting, moreover, that the main attention of the majority of mathematicians was attracted by the concept of the measure of a domain, whereas in the work of the creator of the theory of domains, Cantor, it always occupies a secondary place.

13. In the same volume of M. A.¹ Cantor states an important theorem unifying the theorems $12^\circ$:

B. If $P^{(\alpha)} = 0$, then $P'$, and consequently $P$, are countable, and conversely, where $\alpha$ is an arbitrary number of the first or second class. Then, raising the question of the necessary and sufficient conditions by which the continuum is determined, he classifies domains according to the size of their first derivatives $P'$: the first class consists of those for which $P'$ is countable and consequently $P^{(\alpha)} = 0$; for those $P$ for which $P'$ is uncountable, it can² be decomposed in a unique way into two parts

\[ P' = R + S, \tag{2} \]

where for some number $\alpha$ of the first or second class

\[ R^{(\alpha)} = 0, \qquad S = S' = S^{(\alpha)}; \tag{3} \]

the first domain he called reducible and the second—perfect.

Then Cantor defines the continuum as a perfect and connected domain.

Reprinting the same article in A. M. 2, Cantor, in accordance with Bendixson’s indication, changes his last theorem, namely—the condition satisfied by $R$; in the new exposition $R$ is defined by the condition

\[ D\{R, R^{(\alpha)}\} = 0. \]

Finally Cantor makes several further remarks; he says that

A. a perfect domain is uncountable,

gives an example of a nowhere dense domain

\[ z = \frac{c_1}{3} + \frac{c_2}{3^2} + \cdots + \frac{c_n}{3^n} + \cdots,\quad c_i = 0 \text{ or } 2; \]

it is not difficult to see³ that the specification of this domain is nothing other than the process applied by Veltmann.

Cantor goes on to speak of open domains, an open segment or circle, which he calls semicontinua; in general, a semicontinuum is a connected imperfect domain, any two points of which can be joined by a perfect continuum belonging to the domain.

14. Bendixson (1883) proves,¹) first of all, the theorem:

A. “If $D(P,P') = P$, then $P' \equiv P''$,”

i.e. $P'$ is perfect, and then gives an example of a domain refuting assertion (3) of Cantor. Bendixson here defines the notion of the symmetric placement of an interval $(a,b)$ in an interval $(\alpha,\beta)$ by the equality

\[ a-\alpha=\beta-b. \]

Having constructed symmetrically on $(0,1)$ an interval $\frac12$, then on $\left(0,\frac14\right)$ and $\left(\frac34,1\right)$ two intervals of $\frac18$ each, then four intervals of $\frac1{32}$ each, etc., let us call the set of the endpoints of these intervals the $Q$-domain.

It is obvious that every point $q$ belongs to $Q'$; consequently—by theorem A—the derivative $Q'$ will be perfect.

In each of the free intervals $(\alpha,\beta)$ of the domain $Q$ we place domains $\{x_i\} \equiv P_{(\alpha\beta)}$, constructed in the following manner:

diagram of intervals and points

where $\beta-\alpha=h$ and where $x$ denotes only the interior points of the interval $(\alpha,\beta)$. Obviously,

\[ P'_{\alpha\beta}=(\alpha,\beta). \]

In each of the intervals of the domain $P_{(\alpha\beta)}$ we place domains of the same type; the totality of a countable series of such domains we shall call $P_{((\alpha\beta))}\equiv\{x_{ij}\}$; obviously,

\[ P'_{((\alpha\beta))}=P_{(\alpha\beta)}+(\alpha,\beta),\qquad P''_{((\alpha\beta))}=(\alpha,\beta). \]

Having placed the domains $P_{((\alpha\beta))}$ in each of the free intervals $(\alpha,\beta)$ of the domain $Q$, and having called

\[ \sum P_{((\alpha\beta))}\equiv P, \tag{4} \]

consider its derivatives; it is not difficult to see that

\[ P'=\sum P_{(\alpha\beta)}+Q',\qquad P''=Q'. \]

¹) A. M. 2.

Thus $P'$ is divided into two parts—the perfect set $Q'$ and the countable

\[ R=\sum P^{(\alpha_i)}, \]

where

\[ R'=Q'=Q^{(\Omega)}\ne 0, \]

which refutes Cantor’s theorem. It is evident, however, that $R$ satisfies the condition

\[ D(R,R')=0 \]

or, in general,

\[ D(R,R^{(\alpha)})=0. \]

Further, Bendixson proves the theorems:

D. If $P'$ is uncountable, there exist points belonging to all $P^{(\alpha)}$.

E. The set $P^{(\Omega)}$ of such points is perfect.

F. $P' - P^{(\Omega)} = R$ is countable.

These three theorems were derived simultaneously and independently of one another by Cantor and Bendixson; finally, the following theorem belongs exclusively to Bendixson:

G. There exists a number of the first or second class such that

\[ D(R,R^{(\alpha)})=0. \]

In this way $P'$ can always be brought to the form

\[ P'=R+P^{(\Omega)}, \]

where $P'^{\alpha}=P^{(\Omega)}$ is perfect, and the points $R^{(\alpha)}$, which belong to $P^{(\alpha)}$ and are distinct from the points $R$, must necessarily belong to $P^{(\Omega)}$.

Thus for $P'$ there exists a number $\alpha$ such that

\[ P'=R+P^{\alpha},\quad \text{where } \begin{cases} P^{\alpha}\ \text{is perfect},\\ R^{(\alpha)}=D(P^{(\alpha)}). \end{cases} \tag{5} \]

This equality will hold for any $P'$: for uncountable $P'$—by virtue of the preceding, and for countable ones, because $P^{(\alpha)}=0$.

Apart from the fact that theorem (2) is of capital importance in the theory of sets, and that—consequently—Bendixson’s observation concerning its formulation appears extremely valuable and essential, the very specification of the set (4) is of enormous significance, since it

embodied in concrete forms Cantor’s rather abstract arguments, and can serve as the starting point for a general theory¹ of structure.

Finally, Bendixson makes yet another important observation: the formation of successive derived sets is analogous to differentiation, and the theory of derivatives is analogous to differential calculus; Bendixson lays the foundation for a theory, not yet developed, analogous to integral calculus; he gives the theorem:

For a perfectly rarely scattered set $P$ it is possible to determine infinitely many isolated sets $Q$ such that

\[ Q' = P. \]

Bendixson proves this theorem by basing his argument on the free intervals of the given perfect set.

15. In the Atti della Accademia dei Lincei in 1883, Ascoli published an article, to which he later made additions² in 1888.

In it, considering sets $P$ of points on a line, their limiting points, and the sets $P'$ of these limiting points, Ascoli does not mention in a single word that these concepts owe their origin to Cantor. The difference between Ascoli’s ideas and Cantor’s is only that Ascoli allows that one and the same value $x$ may occur a finite or even an infinitely large number of times, i.e., one and the same point may be regarded as placed on the line more than once. A point of infinite multiplicity eo ipso is a limiting point and enters into $P'$ already as an ordinary point; thus all points of the set $P'$ will be ordinary.

Ascoli further develops Cantor’s ideas, applying them to systems of lines which, like points, may have multiplicity and even infinite multiplicity.

Ascoli establishes the existence of limiting lines, to which a certain variable set of lines tends, in such a way that in an arbitrary neighborhood of each of its points there are points of the curves of the set.

Considering, for a given set of lines, the totality of all limiting lines, Ascoli calls it the first derived set; then he forms derived sets of the second, . . . , $n$-th order and divides sets of lines into two classes according as there exists a finite or an infinite sequence of derived sets.

Every line of infinite multiplicity will necessarily enter into the first derivative as a simple one.

Then Ascoli passes to the notion of the measure of a plane domain; he says that “the totality (insieme) of areas, to each of which there belongs at least one point of a bounded number of lines in the plane $A$, may be made arbitrarily small,” and then: “the sum of such areas for a domain of finite order may be arbitrarily small,” whereby he proves this theorem by applying Cantor’s usual device, and again without mentioning him by a single word.

If the domain of lines $R$ is such that, in the presence of a sequence of infinitely decreasing numbers

\[ \varepsilon_1,\ \varepsilon_2,\ \varepsilon_3,\ \ldots,\ \varepsilon_n,\ \ldots, \]

the number of lines whose projections on the coordinate axes are equal to or greater than $\varepsilon_n$, is finite, then there can be no limiting lines for $R$; for it only limiting points are obtained, so that $R'$ will be only a point domain.

16. In 1884 there was printed $^{1}$ an extract from a letter of Cantor to Mittag-Leffler; in a note to it Mittag-Leffler says that Bendixson, at Cantor’s suggestion, had solved the question of the measure of a perfect domain independently of Cantor and had made a communication in the seminar of Stockholm University.

In M. A. 23 Cantor proves a series of theorems stated by him still earlier, among them the theorems:

C. If $P'$ is countable, then $P^{(\alpha)} = 0$ for some least $\alpha$.

D-E. If $P'$ is uncountable, then there exists a perfect domain $P^{(2)}$, which is identical with $P^{(2)}$,

noting, incidentally, that “if $P^{(\beta)} = 0$, and $\beta$—is a number of the second kind, then $P^{(\alpha)} = 0$, where $\alpha$ is some number of the first kind.”

Here a closed domain is defined by the equality

\[ D(P, P') = P'; \]

such a closed domain can be formed from any domain $P$, by taking $M(P, P')$.

Cantor further gives the theorem: “Every first derivative $P'$ of another domain will be closed, and conversely—every closed domain can be represented in an infinite number of ways as

$^{1}$ A M. 4, p. 381.

derivative of another domain.” Here Cantor again agrees in his conclusions with Bendixson.

Since $Q'$ can always be brought to the form $R+S$, and every closed domain $P$ can be represented as the derivative of some domain $Q$

\[ P \equiv Q', \]

then for a closed domain the theorems C, D, E hold, i.e.

“If $P$ is closed, then for some number $\alpha$ of the first or second class

\[ P \equiv R+S,\quad S \equiv S^{(\alpha)},\quad D(R,R^{(a)}) \equiv 0”. \]

Then Cantor defines a condensed domain by the equality

\[ D(P,P') \equiv P, \]

so that all points of such a domain will be limiting points.

“For a condensed domain $P' \equiv P''$; indeed, let $P' = P+P_g$; then

\[ P'' \equiv M(P',P_g'),\quad \text{i.e. } P' \equiv D(P''); \]

but in general

\[ P'' \equiv D(P');\quad \text{therefore } P' \equiv P''. \]

If a domain has not a single condensed part, then it is called separated (separirt)¹.

To separated domains belong unions and countable closed domains, and in particular $R \equiv P' - P^{(\Omega)}$; indeed—if this were not so, then it would be

\[ P \equiv P_1+P_2, \]

where $P_2$ is the condensed part of $P$; then

\[ P' \equiv M(P_1',P_2'),\quad \text{where } P_2'' \equiv P_2', \]

i.e. $P'$, entering into $P$, would contain a perfect part in its composition, and $P$ could not be a countable domain. Separated also will be a domain of the first kind

\[ P - D\{P,P^{(\Omega)}\}. \]

It is obvious that a condensed domain may not be dense in any interval, and a domain dense in some

an interval may fail to be dense if it still has isolated points inside this interval; if this is not the case, a set that is nowhere dense will often be dense in itself.

Then Cantor proves that the power of every perfect set is equal to the power of the continuum; hence it follows that a closed set is either countable or has the power of the continuum.

17. Passing¹ to the measure of a set, Cantor says that for discontinuous sets we have a certain non-negative number—the integral

\[ \int \int \cdots \int dx_1\,dx_2 \ldots dx_n, \]

extended over all points of the set: this is nothing other than the volume; but in all other cases, too, we can establish a corresponding concept, which Cantor calls Inhalt or Volumen; the number corresponding to this concept must have a definite meaning and a unique value.

Thus here Cantor for the first time expresses the point of view that the measure of a set is a generalization of the concept of length and volume, but he arrives at this generalization with the aid of a certain function $F(\rho)$, depending on a positive infinitely decreasing variable $\rho$. The course of his reasoning is as follows:

For a set $P$ he first forms $M(P,P')$, then around each point $p$ of such a closed set he describes a complete $n$-dimensional ball of radius $\rho$; the set of points inside and on the boundary of the ball Cantor denotes by $K(\rho)$, and then takes the set of points representing the least multiple of all such balls

\[ M_P\{K(\rho,p)\} = \Pi(\rho). \]

The set $\Pi(\rho)$ consists of a finite number of separate continuous parts; extended over all these parts,

\[ \int_{\Pi(\rho)} \int \cdots \int dx_1\,dx_2 \ldots dx_n = F(\rho) \]

Cantor calls the characteristic function of the set $P$; this function is nothing other than the volume of the part of space occupied by the set $\Pi(\rho)$; the limit of this function

\[ \lim_{\rho \ldots 0} F(\rho) = I(P) \]

is what Cantor calls the Inhalt of the set.

Such a definition is—in the first place—extremely complicated in itself; moreover, it requires a preliminary computation of volume by means of multiple integration, which logically ought to precede it. This definition of measure subsequently found a very simple expression in Lindelöf’s article.

If the domain $P$ consists of two separate parts $P_1$ and $P_2$, then obviously

\[ I(P)=I(P_1)+I(P_2); \]

otherwise this equality may fail to hold.

Further,¹) Cantor proves the very important theorem that

\[ I(P)=I(P'). \]

Hence it follows—in the first place—that

\[ I(P)=I(P^{(\alpha)}), \]

if $\alpha$ is any number of the first or second class, and—in the second place—for reducible domains²)

\[ I(P)=I(P^{(\alpha)})=I(0)=0, \]

and for irreducible ones

\[ I(P)=I(P^{(\alpha)})=I(P^{(\Omega)}). \]

We now arrive at the important result that the definition of measure is always reduced to the measure of perfect domains. As for the latter, their measure may be equal to zero,³) but it may also be different from zero.

The measure of a domain depends entirely on the space in which it is considered; for example, the domain of points of a square with side equal to one has measure equal to $1$ and $0$, respectively, in spaces of two and three dimensions. Cantor further says that, for linear domains, measure is easily defined with the aid of free intervals⁴), but he goes no farther in this direction.

18. In 1884,⁵) Mittag-Leffler, making use of Cantor’s doctrine in the theory of functions of a complex variable, gives several examples of domains of various kinds.

¹) p. 475
²) General theorems 12°
³) Thus—for the domain $|z|'\ 13^\circ$ the measure $I(P)$ is equal to zero.
⁴) A. M. 4, p. 390.
⁵) A. M. 4, pp. 58–59

Take

\[ \mathrm{P}_{n}=\left\{2^{-m_{1}}+2^{-(m_{1}+m_{2})}+\cdots+2^{-(m_{1}+m_{2}+\cdots+m_{n})}\right\}, \]

where the $m_i$, independently of one another, run through all integral positive values; we have a set $\mathrm{P}_{n}$ for which $\mathrm{P}_{n}^{(n)}=\{0\}$.

If

\[ \mathrm{P}=\left\{2^{-\mu}+2^{-(\mu+m_{1})}+2^{-(\mu+m_{1}+m_{2})}+\cdots+2^{-(\mu+m_{1}+m_{2}+\cdots+m_{n})}\right\}, \]

where $\mu$, also independently of the $m_i$, is successively equated to all numbers of the natural series, then $\mathrm{P}^{(\omega)}=\{0\}$.

For the set

\[ \begin{aligned} \{&2^{-m_{1}}+2^{-(m_{1}+m_{2})}+\cdots+2^{-(m_{1}+m_{2}+\cdots+m_{n})} +2^{-(m_{1}+m_{2}+\cdots+m_{n}+\nu)}\\ &+2^{-(m_{1}+m_{2}+\cdots+m_{n}+\nu+\rho_{1})} +\cdots +2^{-(m_{1}+m_{2}+\cdots+m_{n}+\nu+\rho_{1}+\rho_{2}+\cdots+\rho_{\nu})}\} \end{aligned} \]

we obtain $\mathrm{P}^{(\omega+n)}=\{0\}$, and $\mathrm{P}^{2\omega}$ will consist of the single point $0$ for the set

\[ \begin{aligned} \mathrm{P}=\{&2^{-\mu}+2^{-(\mu+m_{1})}+2^{-(\mu+m_{1}+m_{2})}+\cdots+2^{-(\mu+m_{1}+m_{2}+\cdots+m_{n})}\\ &+2^{-(\mu+m_{1}+m_{2}+\cdots+m_{n}+\nu)} +2^{-(\mu+m_{1}+m_{2}+\cdots+m_{n}+\nu+\rho_{1})} +\cdots\\ &+2^{-(\mu+m_{1}+m_{2}+\cdots+m_{n}+\nu+\rho_{1}+\rho_{2}+\cdots+\rho_{\nu})}\}. \end{aligned} \]

In 1884, in two articles¹ by Scheffer, one devoted to the notion of the length of the arc of a curve and the other to the theory of functions, the doctrine of sets plays a large role, but the only new point in them is one lemma, from which it follows that a perfect set may consist exclusively of irrational points.

In addition, Scheffer gives an example of a perfect set: take, in the interval $(0,1)$, all decimal fractions in which the digit 5 does not occur; the free intervals here will have as endpoints

\[ 0,a_{1}a_{2}a_{3}a_{4}\ldots a_{n}49 \quad\text{and}\quad 0,a_{1}a_{2}a_{4}\ldots a_{n}6 \]

Phragmén, in the same volume of Acta Mathematica, gives, for $n$-dimensional space, a proof of Bendixson’s theorem $F^{2}$.²

19. In 1884³ there appeared a work by Stolz devoted to the question of the measure of a set; in this respect the author follows Hankel.

The fundamental interval $(a,b)$ he divides into intervals

\[ \tau_{11},\ \tau_{12},\ \ldots,\ \tau_{1m_1}, \]

which constitute the first system of division; then each of the $\tau_{1i}$ is divided into new parts, constituting the second system with intervals $\{\tau_{2i}\}$, and so on; moreover the law of this division is such that, for an arbitrarily prescribed small $\sigma$,

\[ \tau_{ni}<\sigma \quad \text{for } n>n_0. \]

If in each system we take those intervals $\tau_{n\nu_i}$ on which there lie points of the domain $P$, then

\[ \lim_{n\ldots\infty} s_n=\lim \sum \tau_{n\nu_i}=L \]

is a certain completely determined quantity, independent of the law of division of $(a,b)$ into intervals; Stolz calls this limit the Intervallgrenze.

He applies the same process also to a plane point-domain situated in a finite part of the plane bounded by a simple contour.

Stolz divides this part of the plane into rectilinear cells and, analogously to the preceding, proves the existence of a unique limit $L$ for the sum of the cells containing points of $P$; $L$ is the Flächengrenze of the plane point-domain. The cells are represented in the form of polygons whose greatest chord decreases indefinitely.

Returning in 1897¹ to the same ideas, Stolz, already under the influence of Peano and Jordan, besides the limit $L$, defines also

\[ L'=\lim \sum \tau'_{n\nu_i}, \]

where $\tau'_{n\nu_i}$ are those of the elementary cells all of whose points belong to $P$; Stolz now assigns to $L$ and $L'$ the names äussere und innere Flächenzahl. In his “Grundzüge” Stolz takes triangles as elements of area². What is characteristic in Stolz’s articles is the circumstance that he does not adhere to Cantor’s terminology, now generally accepted, and in general mentions him rather little.

20. In 1885 there appeared³ an interesting article by Harnack, devoted to the question of the measure of a domain $P$, i.e.⁴ to the limit of the sum of its

of intervals. If $P$ is dense, then $I(P)=l$; otherwise, in order to determine $I(P)$, Harnack takes the sequence of quantities

\[ \frac{1}{2}l,\ \frac{1}{3}l,\ \ldots,\ \frac{1}{n}l,\ \ldots \]

and constructs intervals equal to or greater than $\dfrac{1}{n}l$ and free of points of $P$. Having selected the first interval $\eta_1$, he applies the same process to each of the remaining parts of $l$, and so on; in general there will be obtained a finite number $\nu$ of free intervals $\eta_i$, equal to or greater than $\dfrac{l}{n}$; let their sum be

\[ \sum_{1}^{\nu} \eta_i = s_n \quad \text{for } \eta_i \geqq \frac{l}{n}. \]

The points of the domain $P$ which lie outside these intervals or—in the extreme case—serve as their boundaries are situated in a likewise finite number of intervals $\{\varepsilon_i\}$, whose sum is equal to $l-s_n$.

Consequently—the points of the domain may be included in a finite number of intervals whose total length differs arbitrarily little from $l-s_n$; for this it is only necessary to diminish the intervals $\eta_i$ arbitrarily little. The limit

\[ \lim_{n\ldots\infty}(l-s_n)=s \]

is the sought Intervallgrenze.

On the basis of this definition, $l-s_n$ will include not only those intervals all of whose points are points of $P$, but also those in which the points of $P$ are only dense, irrespective of whether they are countable or uncountable.

In Harnack’s definition two points are extremely important: first, it is based on intervals free of points of $P$, the arrangement of which is determined by the very nature of the domain $P$; and second, it indicates the possibility of including the points $P$ in a finite number of intervals whose sum is arbitrarily close to $s$. Both of these circumstances, although in particular cases they had also appeared in Cantor and du Bois-Reymond, are essential; many years after Harnack, Osgood applied them in this respect.¹

Next, Harnack is a predecessor of Peano and Jordan in the sense of the definition of the inner and outer measure of a domain, although here he somewhat contradicts his own definition.

The matter is as follows: the limiting points of a domain $P$ need not be included in this domain; it is possible that these points, constituting the domain

\[ P' - D(P,P') = T, \]

are such that the intervals containing them—according to Harnack’s definition—may have a sum $s_1$ not equal to zero; let, on the other hand, the sum of the intervals occupied entirely$^1$ by the points $P$ be $s$. Then the sum of the free intervals must be $l - s - s_1$; thus, in Harnack’s words, “the measure of the domain $P$ will be $s + s_1$,” despite the fact that $P$ can be included in an infinite series of intervals with total length $s$.

The assertion made here is not quite exact: since in every interval containing the points $T$ there must also be points $P$, the measure of the domain, as Harnack defines it, will necessarily be $s+s_1$; in this he is perfectly right, but that $P$ can be included in a series of intervals with total length $s$ is not true. In later terminology, $s$ will be the inner, and $s+s_1$ the outer measure of the domain.

Next, Harnack was the first to express$^2$ the idea that every, even densely scattered, countable domain has the property that its points can be included in intervals whose sum is arbitrarily small; in the case of a densely scattered domain, concerning the points “which are not covered by these intervals,” the author observes$^3$ that they will not be densely scattered in any interval, but will have measure greater than $1-\delta$.

21. Further, Harnack raises, as Stolz did, the question whether the process of determining measure is single-valued, i.e. whether the same number will be obtained as the measure of the domain if the choice of the intervals free of points is made according to a different law than before. This question is especially not superfluous for plane multidimensional domains, where the selection of spheres free of points of the domain can be made in very diverse ways.

$^1$ Only in this sense can Harnack be understood.
$^2$ P. 242.
$^3$ P. 243.

For a linear domain, Harnack argues as follows: let two different processes have successively given us sums of intervals

\[ \tag{1} s_1,\ s_2,\ s_3,\ldots,\ s_m,\ldots, \]

\[ \tag{2} t_1,\ t_2,\ t_3,\ldots,\ t_n,\ldots, \]

obtained after the removal of the free intervals. Since $s_i$ and $t_i$ do not increase, there exist for them the limits

\[ \lim s_m=\sigma,\qquad \lim t_n=\tau. \]

Let us take some term $s_m$ from the series (1) such that

\[ s_m=\sigma+\delta, \]

where $\delta$ is some small positive number.

If we take any system of intervals from the series (2), then these intervals will in part lie on $s_m$, and in part will be outside $s_m$; since this latter part is not included in $s_m$, there are no points of $P$ on it, and it must fall away in the course of the subsequent process; thus we have the possibility of finding such a term $t_n$ of the series (2) that the part of $t_n$ extending beyond the limits of $s_m$ will be less than some small quantity $\varepsilon$; in such a case it will turn out that

\[ t_n<s_m+\varepsilon. \]

Arguing similarly with respect to $t_n$, we can find such a sum $s_{m'}$ from the series (1) that

\[ s_{m'}<t_n+\varepsilon. \]

Therefore we have

\[ s_{m'}-\varepsilon<t_n<s_m+\varepsilon, \]

whence it follows that

\[ \sigma=\tau. \]

The process proposed by Harnack for determining the measure of two-dimensional domains differs essentially from the preceding one and is not its development.

Let us imagine a plane domain enclosed in a circle of some radius $l$; let us divide the interior of the circle into such parts that would satisfy the following conditions: $a)$ their areas can be determined by the ordinary methods of integration, $b)$ these areas admit arbitrary diminution, while $c)$ the greatest distance between points of these areas decreases without bound.

Let the areas of the first system of division be less than $\delta$; let us remove those of them in whose interior or on whose boundary there are no points of the domain. We divide the remaining areas into parts smaller than $\delta'$, with $\delta' < \delta$; from them we again remove the areas free of points of the domain; and so on. The areas remaining after such a process give non-increasing sums

\[ s_1, s_2, s_3, \ldots, s_n, \ldots, \]

for which there exists a limit

\[ \lim s_n = s, \]

independent of the law of division.

The step backward in Harnack’s reasoning must be seen in the respect that here he does not determine the free intervals determined by the structure of the domain, but divides the basic interval into parts not standing in any relation to this structure. That these two processes can lead to entirely different results was indicated by Borel¹.

Thus Harnack’s definition of the measure of a plane domain proceeds from quite different foundations than his own definition of the measure of a linear domain. Moreover, the author has here to rely, in computing the areas of curvilinear figures, i.e., on the methods of integral calculus, whereas logically the definition of the measure of a domain should precede the computation of areas, which is its generalization, and should therefore be constructed independently of integration; and this is all the more so since² “the area of a plane curvilinear figure, as a quantity, on a par with the length of an arc, is precisely one of those geometrical magnitudes which our mind clearly understands, or thinks that it understands, but which must be exactly defined before introducing them into analysis; and this applies especially to the concept of area, since other proofs are usually based upon it.”

Finally, one must note in Harnack a second example, after Veltmann, of the construction of a plane point domain.

Taking a linear point domain on the axis of abscissas and erecting at each of its points a perpendicular equal to unity, we compose a domain from all the points of all the perpendiculars. This plane domain has, as its measure, the measure of the linear domain. In this connection the author

observes that, if a linear set is nowhere dense, then a plane set whose measure may be non-zero has the particular property that it contains no interior points.

Harnack’s idea can be used to construct plane sets of a previously prescribed character: taking—for example—given linear sets on both axes and forming the plane set from the points of intersection of the corresponding perpendiculars to the axes, one can obtain different sets depending on the character of the linear sets.

22. In 1885¹ Cantor, summing up the results of his preceding works,² gives several consequences that had not earlier been proved; namely, he observes “that every perfect set is closed and dense-in-itself,” and that “$P^{(z)}$ is closed.”

The author then sets himself the aim of generalizing the preceding theorem³ $P = R + S$ for a closed set, by establishing it for arbitrary sets, which may or may not be closed, and by first defining the concept of homogeneity. If, about the points of the dense-in-itself part of a set $P$, one describes a sufficiently small sphere, and if the measure of the set of points of $P$ lying in it is always one and the same, then the set $P$ is called homogeneous.

Taking an arbitrary set $P$, Cantor divides it into two parts

\[ P = P_a + P_c, \]

where $P_a$ is the set of isolated points and $P_c$ the set of limit points; he calls $P_a$ the Adhärenz and $P_c$ the Cohärenz of the set $P$; moreover, every dense-in-itself part of $P$ is contained in $P_c$. The same process may be applied to $P_c$, yielding

\[ P_c = P_{ca} + P_c^2,\qquad P = P_a + P_{ca} + P_c^2, \]

where $P_{ca}$ contains all isolated points of $P_c$. In general,

\[ P = \sum_{z' < z} P_c^{z'} a + P_c^z \]

and then

\[ \tag{1} P = \sum_{z < \Omega} P_c^z a + P_c^\Omega; \]

where $x$ is an arbitrary number of the first or second class, and $\Omega$ is the first number of the third class. Here each of the $P_c^{x}{}_a$ is isolated, and $\sum P_c^{x}{}_a$ is separable.

“If $P$ is separable, then $P_c^{x} = 0$ for some least number $x$ of the first or second class; if $P$ is not separable, then $P_c^{x}$ is dense.”

Indeed:

A. Let $P$ be countable; then the sum

\[ \sum_{x<\Omega} P_c^{x}{}_a \]

must consist of a countable sequence of elements; consequently there must be some least $x$ for which

\[ P_c^{x}{}_a = 0,\qquad P_c^{x} = P_c^{x}{}_a + P_c^{x+1} = P_c^{x+1} = P_c^{x+\lambda}; \]

if $P_c^{x} \ne 0$, then $P_c^{x}$, and consequently $P_c^{x+\lambda}$, will be dense, and $P$ is not separable; but if $P$ is separable, then $P_c^{x} = 0$.

B. If $P$ is uncountable, then in the continuous interval in which $P$ is situated there are points $\{q\}$, in every neighborhood of which lies an uncountable part of $P$; the domain $Q$ of these points must be closed, and moreover

\[ D(P,Q)=V\ne 0; \]

the points $V$ cannot be isolated, since $V$ will be a dense domain.

Thus every uncountable domain contains within itself a dense part. Hence it follows that every separable domain is countable.

Since in (1)

\[ \sum_{x<\Omega} P_c^{x}{}_a = R \]

is always separable, it will be countable; and in that case, for some least $x$,

\[ P_c^{x}{}_a = 0,\qquad P_c^{x} = P_c^{x+1} = P_c^{\Omega}, \]

whereas $P_c^{x}$ for an uncountable domain is not equal to zero, since $V$ constitutes part of $P_c^{\Omega}$.

If we call $U$ the domain of the points $P_c^{(2)}$ not included in $V$,

\[ P_c^{\alpha} - P_c^{(2)} = U + V, \]

then $U$ may be zero or will be a homogeneous domain of the first order.

Thus every domain can be decomposed into the following component parts

\[ P = R + U + V. \]

For a closed domain

\[ P_c = P', \qquad P_a = P - P' \]

\[ P_c^{\alpha} = P^{(\alpha)}, \qquad P_{c\,a}^{\alpha} = P^{(\alpha)} - P^{(\alpha+1)}, \qquad P_c^{(2)} = V = P^{(\Omega)}, \]

since here $P^{(\Omega)}$ is zero or perfect, and consequently $U = 0$.

Cantor further admits the existence of domains of rank higher than the second and, in accordance with this, divides $V$ into homogeneous parts of higher orders.

Finally, the author gives a classification of limiting points: the isolated points of the domain $P$, constituting $P_a$, are called points of the 0-th genus; the isolated points $P_c$ which yield $P_{c\,a}$ are called points of the first genus, and so on; in general, the isolated points $P_c^{\alpha}$ forming the domain $P_{c\,a}^{\alpha}$ are called points of the $\alpha$-th genus.

In accordance with this, the points belonging to $P_c^{(2)}$ are divided into orders. All points of the $\alpha$-th genus yield an isolated domain; all points of the $\beta$-th order yield a homogeneous domain of the $\beta$-th order. From the absence of points of genus $\alpha$ follows the absence of points of genus $\alpha + \lambda^0$; whereas for points of different orders no such interdependence exists. All points of the domain $P$ will be limiting points of the $\alpha$-th genus or of the $\beta$-th order, except for the points $P_a$, which are isolated.

Strange as it may be, Schönflies in his “Bericht” did not say a single word about this classification of the points of a domain, which is very important from the point of view of the structure of the domain.

The considerations set forth above concerning the structure of a domain are very essential for clarifying it. But, notwithstanding this, they have until now remained wholly untouched by any other authors; the reason—perhaps—to some extent was the extreme abstractness of all the reasoning.

The second chapter of the present investigation is closely connected with the preceding one, although it proceeds from different considerations.

23. In 1887, Volterra¹ defines a neighborhood of a curve in a space $L$, which is assumed to be closed or to extend to the boundary of a domain, if it belongs to a domain bounded by some surface; moreover, $L$ has no multiple points and admits a tangent everywhere except at a finite series of points.

Let, for $L$, some closed curve $C$ be taken, concatenated (concatenata) with $L$, i.e., such that $L$ passes inside $C$.

Then, by moving $C$ without breaking the concatenation, we obtain a certain tubular surface inside which $L$ will lie. The points inside this surface will be a neighborhood of the line $L$ (interno della linea). By taking $C$ sufficiently small, we can obtain an arbitrarily small neighborhood of $L$.

Since we are dealing with the plane or with space, the extension of the concept of a neighborhood becomes essentially necessary; therefore Volterra’s definition is important for the theory of multidimensional domains.

In 1889, Arzelà², developing Volterra’s ideas, considers, in the plane, as elements, points and lines for which limit elements can be defined.

A limit element is one such that, in an arbitrary neighborhood of it bounded by lines whose distance from it is finite (maggiore di un numero assegnabile), there are infinitely many other elements; at the same time, from each of them all points of the limit element are at a distance less than an arbitrarily small $\varepsilon$.

For infinitely many points there always exists a limit point; the question arises whether the same holds for an infinite set of lines. Arzelà proves that under certain conditions this is indeed realized.

24. In 1887, Peano establishes³ interesting concepts.

Let $P$ be a linear domain (campo di punti); for it, $x$ will be an interior point if it is possible to determine a neighborhood $(x-\rho, x+\rho)$ such that all its points will be points of $P$; $x$ will be an exterior point if all points of its neighborhood $(x-\rho, x+\rho)$ do not belong to $P$; points satisfying neither the one nor the other condition will be boundary points (punto limite); the latter points always exist if the domain does not comprise all points of the line; the domain of these points is the campo limite.

The points of a straight line between two given points, including these latter points or not, form a domain that is called a rectilinear segment; its length is its “principal magnitude”; every domain consisting of a finite number of segments likewise has a length comparable with the length of a rectilinear segment.

For an arbitrary domain $P$ we may imagine domains $\Pi$ and $\bar{\pi}$, each consisting of a finite number of segments, of which the first contains $P$ as a constituent part, while the second, on the contrary, is contained in $P$; $\Pi$ and $\bar{\pi}$ have lengths $M(\Pi)$ and $M(\bar{\pi})$, and $M(\Pi)$ is not less than $M(\bar{\pi})$.

If, for variable $\Pi$ and $\bar{\pi}$, the lower limit of $M(\Pi)$ coincides with the upper limit of $M(\bar{\pi})$, which we shall denote$^1$ by

\[ \mathrm{uGr}\, M(\Pi) = \mathrm{oGr}\, M(\bar{\pi}), \]

then their common value is called the length of the rectilinear domain.

If these limits are not equal, the domain $P$ has no length comparable with the length of a segment; then $\mathrm{uGr}\, M(\Pi)$ and $\mathrm{oGr}\, M(\bar{\pi})$ Peano calls respectively the outer and inner length of the domain.

In the special case, if there is no domain $\Pi$ enclosing $P$, the outer length of $P$ is equal to infinity; if there is no domain $\bar{\pi}$ entering into the composition of $P$, the inner length of $P$ is equal to zero.

The domain of boundary points enters into the composition of the domain $\Pi - \bar{\pi}$, which also consists of a finite number of segments, and moreover

\[ M(\Pi - \bar{\pi}) = M(\Pi) - M(\bar{\pi}) \]

is the outer length of the domain of boundary points.

Peano establishes the same concepts for domains of two and three dimensions, while the limite itself he now calls a contour (contorno); the domains $\Pi$ and $\bar{\pi}$ here consist of polygons and prismatic bodies.

Peano’s ideas found further development in the works of Jordan.

25. In 1892,$^2$ Jordan lays at the foundation of the whole analysis the theory of domains.

Into this theory he introduces certain new concepts necessary for its extension to multidimensional domains. He calls the distance (écart) between two given points the expression

\[ \rho = |x_1 - x_2| + |y_1 - y_2| + \cdots + |t_1 - t_2|. \]

$^1$ Here $\mathrm{uGr}$ and $\mathrm{oGr}$ are abbreviations of “untere Grenze” and “obere Grenze.”

$^2$ Journal de Mathématiques, (4) 8, and in 1893 — Cours d’Analyse, deuxième édition, t. I.

introduced by him in place of distance

\[ r^2=(x_1-x_2)^2+(y_1-y_2)^2+\cdots+(t_1-t_2)^2; \]

the greatest separation of the points of a domain is its diameter. The lower bound of the separations of points belonging respectively to two domains is the separation of the domains.

For a given domain $P$, not containing all possible points of the known space, all points not entering into $P$ constitute the complementary domain $\Pi$. Jordan then defines¹ the interior and exterior points of a domain and the points of its contour (frontière).

The contour always represents a closed domain, because every one of its limit points will be a point of the contour.

Then Jordan, like Peano, defines $E(P)$ and $e(P)$—the inner and outer extent (étendue), taking as a basis two systems of parallel straight lines and proving that the system of divisions does not influence the result.

If $Q$ is a domain interior with respect to $P$, then

\[ e(Q)<E(Q)<e(P); \]

if

\[ P=\sum_1^n P_i, \]

then

\[ E(P)<\sum_1^n E(P_i), \qquad e(P)>\sum_1^n e(P_i); \]

if

\[ E(P)=e(P), \]

then the domain is called measurable (mesurable).

26. In view of the circumstance that, in the second chapter of the present investigation, as the fundamental idea in the construction of a domain there are taken the well-known types of arrangement, we shall have to set forth what Cantor understands by the same term, and thus to clarify what is common between our general method of constructing domains and Cantor’s theory, systematically expounded in 1895–97.²

A domain is called simply arranged (einfach geordnete) if among the elements there exists such an order that $a)$ of two elements one occupies the lower and the other the higher place, and $b)$ of

of three elements, if the first precedes the second, and the second the third, then the first precedes the third.

To every simply ordered domain there corresponds a definite type of ordering (Ordnungstypus) of its elements; by this Cantor understands the general concept which arises if we abstract from the individual properties of the elements of the domain, but retain only their order.

For every transfinite cardinal number there is an infinite multitude of different types of ordering, constituting a special class of types. The simplest infinite types will be $\omega$ and ${}^{*}\omega$, corresponding to the orderings of, respectively, the positive integers and the negative integers

\[ \ldots,\,-n,\,\ldots,\,-3,\,-2,\,-1,\,0,\,1,\,2,\,3,\,\ldots,\,n,\,\ldots; \]

the type of ordering of all integers is ${}^{*}\omega+\omega$.

Cantor performs addition and multiplication on types of ordering, and for addition in general

\[ \alpha+\beta \ne \beta+\alpha,\qquad (\alpha+\beta)+\gamma=\alpha+(\beta+\gamma); \]

if $n$ is a finite type of ordering, then—among other things—

\[ n+\omega=\omega,\qquad \omega+n-\omega; \]

then

\[ \alpha\beta \ne \beta\alpha,\qquad \alpha\beta\cdot\gamma=\alpha\cdot\beta\gamma,\qquad \alpha(\beta+\gamma)=\alpha\beta+\alpha\gamma. \]

Among the various types of ordering Cantor considers, besides a) $\omega$ and b) ${}^{*}\omega$, also c) $\eta$—the type of ordering of the positive rational proper fractions in their natural order—and d) the type of the linear continuum. Besides these, one may also take the types of ordering e) of a dense, f) of a closed, and g) of a perfect domain.

27. In 1895¹, in his dissertation, Borel establishes a theorem which is now known as “Borel’s theorem”:

“If on a finite interval there is a countable² sequence of intervals of such a kind that every point of the interval is an interior point of at least one of the intervals, then some finite number of these intervals already covers the interval entirely, i.e. in such a way that all its points will be interior points of this finite sequence of intervals.”

Indeed: let $a$ be the right end of the whole interval, and $(a_1,b_1)$ one of the intervals containing $a$ within itself; $(a_2,b_2)$ one of the intervals containing $b_1$, and so on; these intervals either reach the right end $b$ of the whole interval, or they do not, and then there exists

\[ b_\omega=\lim_{i.\infty} b_i . \]

For $b_\omega$ we find the corresponding interval $(a_{\omega+1}, b_{\omega+1})$, and let $a_{\omega+1}$ fall between $b_{m-1}$ and $b_m$. In that case it is enough to retain a finite number of intervals $(a_1,b_1), (a_2,b_2), \ldots, (a_m,b_m)$, $(a_{\omega+1}, b_{\omega+1})$, in order to get from $a$ to $b_\omega$, and so on. We always have the possibility of continuing this process, passing, when necessary, to the limit and thereby replacing a denumerable sequence of intervals by a finite one; in this way we shall reach the end $b$, because the process of finding the intervals $(a_i,b_i)$ must break off; otherwise we should obtain a sequence of intervals with endpoints

\[ b_1,\ b_2,\ b_3,\ldots,\ b_\omega,\ldots, \]

where the indices run through all numbers of the second class. But these indices, since the domain of the intervals is denumerable, are natural numbers in their natural order, and this is impossible, since the second class of numbers forms a domain of the second cardinality.

We thus arrive—in accordance with the preceding process—at the definition of a division into a finite number of intervals covering a segment of the straight line.

In 1898$^1$ Borel gives another proof of his theorem, which can hardly be regarded as satisfactory: it contains a petitio principii. Having renumbered the intervals in some order and wishing to prove that, after some $N$ intervals, there will remain no points that have not entered into at least one of them, Borel assumes the existence of at least one such point; then the intervals which contain it have indices exceeding any prescribed number $n$. Borel divides the whole segment in halves, then again in halves, and so on, and takes those successive parts of the original interval which contain one another and which will all have the property inherent in the whole segment, namely—each of the successive parts contains at least one point that belongs only to intervals with indices greater than an arbitrary number $n$.

$^1$ Leçons, pp. 42–43.

The decreasing parts of the interval, of lengths

\[ l,\quad \frac{l}{2},\quad \frac{l}{2^2},\ldots,\frac{l}{2^m},\ldots \]

have as their limit a certain point $\alpha$, which, as Borel says, “by assumption lies inside a definite interval $(a_k,b_k)$, since we have assumed the domain of intervals to be countable.” In these words Borel appeals to what still has to be proved. The reason which entails the failure of the proof lies in arranging the intervals in a definite order.

A simpler proof of Borel’s theorem was given by Lebesgue¹.

If from the left boundary $a$ of the whole interval $(a,b)$ one can, by means of a finite number of intervals, reach the point $x$, Lebesgue calls this point attained. If $x$ is attained, then every point of the interval $(a,x)$ is also attained; in the contrary case all points of $(x,b)$ will not be attained. If $b$ is not attained, there is a last attained or first unattained point $x_0$; this point lies inside some interval $(\alpha_0,\beta_0)$; let

\[ \alpha_0 < x_1 < x_0 < x_2 < \beta_0 . \]

By the assumption concerning $x_0$, the point $x_1$ is attained after a finite number $n$ of intervals; adding to them the interval $(\alpha_0,\beta_0)$, we attain the point $x_2$, which contradicts the position concerning the role of the point $x_0$; thus $b$ must be attained. Here, as Lebesgue observes, there is no restriction that the number of all intervals be countable. Lebesgue gives an analogous proof also for a two-dimensional domain, making use, in doing so, of the Peano curve.

28. Since in the countable sequence of intervals figuring in Borel’s theorem there are intervals which cover one another, the sum of all intervals

\[ \sum_1^\infty l_n \]

will exceed the length of the basic interval $l$. The same happens also with respect to the intervals $l', l'',\ldots,l^{(N)}$, which cover the whole basic interval and exist by virtue of the preceding theorem:

\[ \sum_{j=1}^{N} l^{(j)} > l . \]

It follows directly from this that if the sum of a finite or infinite series of intervals is less than $l$, then outside these intervals there must exist points of the fundamental interval.

These points will be non-denumerable; indeed: let

\[ \sum_{1}^{\infty} l_n = s < l, \]

and let the exterior points form a denumerable series

\[ \tag{1} a_1,\ a_2,\ a_3,\ldots,\ a_m,\ldots; \]

construct around them the intervals

\[ l'_m=\left(a_m-\frac{\varepsilon}{2^m},\quad a_m+\frac{\varepsilon}{2^m}\right), \]

then we shall have two denumerable series of intervals $\{l_n\}$ and $\{l'_m\}$, containing all the points of the fundamental interval $l$. The sum of these intervals, taken together, will be no greater than $s+\varepsilon$; since $\varepsilon$ can be chosen so as to satisfy the condition $s+\varepsilon<l$, it follows from this that outside $l_n$ and $l'_m$ there must be points of $l$, contrary to the assumption. Thus the exterior points cannot form a denumerable series.

From the preceding there follows, among other things, a very important theorem, which Borel touched upon only in passing: “Every denumerable domain can be enclosed in intervals whose total sum can be made arbitrarily small.”

29. Taking the intervals

\[ \tag{2} \left(\frac{p}{q}-\frac{1}{q^3},\quad \frac{p}{q}+\frac{1}{q^3}\right), \]

surrounding all rational points, Borel considers the domain of points exterior to these intervals; it is of interest to examine such a domain in greater detail.

Writing the rational fractions in the form of a denumerable series

\[ \frac12;\ \frac13,\ \frac23;\ \frac14,\ \frac34;\ \frac15,\ \frac25,\ \frac35,\ \frac45;\ \frac16,\ \frac56;\ \frac17,\ \frac27,\ \frac37,\ \frac47,\ \frac57,\ \frac67;\ldots, \]

we obtain for them the corresponding intervals

\[ l_1=\left(\frac38,\ \frac58\right),\quad l_2=\left(\frac{8}{27},\ \frac{10}{27}\right),\quad l_3=\left(\frac{17}{27},\ \frac{19}{27}\right),\quad l_4=\left(\frac{15}{64},\ \frac{17}{64}\right),\quad l_5=\left(\frac{47}{64},\ \frac{49}{64}\right), \]

\[ l_6=\left(\frac{24}{125},\ \frac{26}{125}\right),\quad l_7=\left(\frac{49}{125},\ \frac{51}{125}\right),\ldots . \]

It is not difficult to see that 1) some of these intervals are placed entirely on others; to this category belongs, for example, $l_2$, lying on $l_1$, since

\[ \frac{3}{8}=\frac{375}{1000}<\frac{392}{1000}=\frac{49}{125},\qquad \frac{51}{125}=\frac{408}{1000}<\frac{625}{1000}=\frac{5}{8}; \]

[Diagram of the intervals on a line, with points marked $A_0, A^{\mathrm{II}}, A^{\mathrm{I}}, A, B, B^{\mathrm{I}}, B^{\mathrm{II}}, B_0$ and intervals $l_1^{\mathrm{I}}, l_1^{\mathrm{II}}, l_2^{\mathrm{I}}, l_2^{\mathrm{II}}$.]

and 2) the midpoint of some intervals coincides with the endpoints of others; indeed, the endpoints of the interval $\left(\frac{3}{8},\frac{5}{8}\right)$ will be rational, and therefore each of them serves as the midpoint of the intervals

\[ \left(\frac{3}{8}-\frac{1}{8^3},\ \frac{3}{8}+\frac{1}{8^3}\right) \quad\text{and}\quad \left(\frac{5}{8}-\frac{1}{8^3},\ \frac{5}{8}+\frac{1}{8^3}\right); \]

in turn, the points

\[ \frac{3}{8}-\frac{1}{8^3}=\frac{319}{2^9},\qquad \frac{5}{8}+\frac{1}{8^3}=\frac{321}{2^9}, \]

serve as midpoints of the intervals

\[ \left(\frac{319}{2^9}-\frac{1}{2^{27}},\ \frac{319}{2^9}+\frac{1}{2^{27}}\right), \quad \left(\frac{321}{2^9}-\frac{1}{2^{27}},\ \frac{321}{2^9}+\frac{1}{2^{27}}\right); \]

and so on; we shall obtain in this way, to the left and to the right of each interval $l=AB$, intervals $l_1^{\mathrm{I}}$ and $l_2^{\mathrm{I}}$ with midpoints $A, B$; intervals $l_1^{\mathrm{II}}$ and $l_2^{\mathrm{II}}$ with midpoints $A^{\mathrm{I}}$ and $B^{\mathrm{I}}$, and so on. These intervals, lying with one half on the previously taken intervals, lengthen the interval with the other half in both directions; the points $A, A^{\mathrm{I}}, A^{\mathrm{II}}, \ldots$ and $B, B^{\mathrm{I}}, B^{\mathrm{II}}, \ldots$, etc., tend to certain limits $A_0$ and $B_0$, which lie outside the expanding interval $A^{(i)}B^{(i)}$ for any value of $i$. These points will be the endpoints of the interval $A_0B_0$, to which $A^{(i)}B^{(i)}$ tends as $i$ increases, and we may therefore regard $A^{(i)}B^{(i)}$, with variable $i$, as the open interval $A_0B_0$.

We have seen that some of the intervals $l_j$ are situated on preceding intervals, so that the corresponding endpoints $A_0, B_0$ may lie inside other intervals. This occurs, for example, for the interval $\left(\frac{49}{125},\frac{51}{125}\right)$; the question is whether this will not be a general phenomenon, i.e., whether all the intervals are not arranged in such a way that they continually cover one another. If this is so, then inside the intervals $l_j$ there must be found all points $(0,1)$ without exception.

To answer this question, we take the sum of all possible intervals, without concern for whether they cover one another or not. Since we are dealing with proper fractions, for each denominator $q$ the numerator may take values from 1 to $q-1$; thus there will be $q-1$ intervals of type (2). Each of them has length $\dfrac{2}{q^3}$, so that all together they give $\dfrac{2(q-1)}{q^3}$. Giving $q$ all possible values from 1 to infinity, we obtain

\[ s=\sum_1^\infty \frac{2(q-1)}{q^3} =2\sum_1^\infty \left(\frac{1}{q^2}-\frac{1}{q^3}\right) =2M<1, \]

since $M$ is a certain definite number $<\dfrac12$. From Borel’s theorem it follows that in this case there exists a non-denumerable domain of points not lying inside the intervals (2).

Thus all the endpoints of the open intervals $A_0B_0$ cannot lie inside other intervals $l_j$; consequently, there will be such endpoints $A_0,B_0$ that do not belong to this category, and they must necessarily be irrational.

In this way we obtain a domain of open intervals of type $A_0B_0$, whose non-interior points will be exclusively irrational points; these points form a perfect set.

Concerning what has been set forth above, Borel says¹: “Having reflected on the fact that one can remove from the line all the points contained in each of the intervals (2), and that there still remains a non-denumerable domain of points, we shall be less inclined to think that we know what continuity is, and to reason about it as about an intuitive and perfectly clear concept.”

30 Further, Borel gives an important indication concerning the nature of discontinuity and making it possible to draw an even more striking conclusion.

Around the rational points $\dfrac{p}{q}$ we construct the intervals

\[ \tag{3} \left(\frac{p}{q}-\frac{\varepsilon}{q^3},\ \frac{p}{q}+\frac{\varepsilon}{q^3}\right), \]

where

\[ \varepsilon=\frac{1}{\nu}\quad \text{for } \nu=1,2,3,4,\ldots . \]

The sum of the intervals $l_n^{(\nu)}$ corresponding to some value $\nu$ satisfies the inequality

\[ \sum_{n=1}^{\infty} l_n^{(\nu)} \leq 2\varepsilon \sum\left\{\frac{1}{q^2}-\frac{1}{q^3}\right\} = 2\varepsilon M - \frac{2M}{\nu}, \]

where this sum may be arbitrarily small in connection with the magnitude $\nu$.

Taking the set of points lying inside the intervals (3), we obtain, depending on the value of $\nu$, a sequence of sets

\[ E_1,\ E_2,\ E_3,\ldots,\ E_\nu,\ldots, \tag{4} \]

each of which is contained in all the preceding ones; let us consider the set of those points which are common to all the sets (4)

\[ E = D\{E_1,E_2,E_3,\ldots,E_\nu,\ldots\}. \]

That all $\frac{p}{q}$ must belong to $E$ is self-evident; but it turns out that, besides these, $E$ also contains an uncountable sequence of certain other points. To see this, take the set of numbers

\[ \xi = \frac{\alpha_1}{10^{1!}}+\frac{\alpha_2}{10^{2!}}+\frac{\alpha_3}{10^{3!}}+\cdots+\frac{\alpha_n}{10^{n!}}+\cdots, \tag{5} \]

where $\alpha_i$ may take any values from the sequence of numbers from 0 to 9.

Since the set (5) may be put in one-to-one correspondence with the set of numbers

\[ x=\frac{\alpha_1}{10^1}+\frac{\alpha_2}{10^2}+\frac{\alpha_3}{10^3}+\cdots+\frac{\alpha_n}{10^n}+\cdots, \]

and these latter give all points¹ of the interval $(0,1)$, which are uncountable, the numbers (5) will likewise be uncountable, and the set $\{\xi\}$ has the cardinality of the continuum.

We shall now prove that each of the numbers $\xi$ belongs to each of the sets $E_\nu$, for an arbitrary value of $\nu$.

Let $\xi$ be one of the numbers (5); representing it in the form

\[ \xi=\frac{p}{q}+\frac{\alpha_{m+1}}{q^{m+1}}+\frac{\alpha_{m+2}}{q^{(m+1)(m+2)}}+\cdots,\quad \text{where } q=10^{m!}, \]

we see that

\[ \tag{6} \left|\xi-\frac{p}{q}\right|<\frac{1}{q^m}; \]

here $\frac{p}{q}$ is one of the approximations to the number $\xi$, since $m$, and consequently also the denominator $q$, may be taken arbitrarily large, so that $\frac{p}{q}$ may differ arbitrarily little from $\xi$.

The numbers belonging to any one of the sets $E_\nu$ satisfy the condition

\[ \tag{7} \left|\alpha-\frac{p}{q}\right|<\frac{1}{\nu}\cdot\frac{1}{q^3}. \]

If $m$ is chosen so that

\[ \tag{8} \frac{1}{q^m}<\frac{1}{\nu}\cdot\frac{1}{q^3},\qquad q^{m-3}=10^{m!(m-3)}>\nu, \]

then the number $\xi$—by virtue of (6)—will satisfy condition (7) and therefore is one of the numbers of the set $E_\nu$. Inequality (8), however, can always be realized by choosing $m$ suitably for any given finite $\nu$.

Thus the uncountable set $\{\xi\}$ enters, as a constituent part, into each of the $E_\nu$; consequently it will also enter into $E$, whence follows the uncountability of the latter set. This fact, established by Borel, is of very great significance.

The intervals (3), by which the sets $E_\nu$ are defined, have rational points as their midpoints and endpoints. The infinite decrease of the intervals always occurs in such a way that the endpoints remain rational; consequently—the set $E$ is defined by a countable sequence of infinitely small intervals with rational endpoints.

From the fact that the set $E$ is uncountable it follows further that in some one of these intervals there is an uncountable set of points.

Indeed, if in each of them there were only a finite or countable number of points, then a countable sequence of intervals would yield a countable sequence of finite or countable sequences of points; and, as is known, such a sequence will again be only countable. Thus at least one of the intervals (3) must possess an uncountable sequence of points.

But it is not difficult to see that, if this holds with respect to one of the intervals, then the same is true with respect to-

tively for all. In fact: for a given value of $\nu$, between every two intervals (3) of the domain $E_\nu$ one can establish a one-to-one correspondence which will be preserved at each stage of the change of $\nu$; it will therefore also be preserved for the intervals of the domain $E$.

We thus arrive at the conclusion that each of the intervals of the domain $E$ must yield an uncountable domain of points of $E$, or—in other words—between any two arbitrarily little separated rational numbers there lies an uncountable domain of other numbers.

31. If we apply the theorem at the end of § 28 to the rational points of the interval $(0,1)$, which form a countable sequence, we arrive at rather interesting results.

Let

\[ x_1, x_2, x_3,\ldots, x_n,\ldots \]

be a countable sequence of all rational fractions; further, let $\varepsilon$ be some small irrational number; construct intervals $l_i$ of lengths

\[ \left(x_1-\frac{\varepsilon}{2},\ x_1+\frac{\varepsilon}{2}\right),\ \left(x_2-\frac{\varepsilon}{2^2},\ x_2+\frac{\varepsilon}{2^2}\right),\ \ldots,\ \left(x_n-\frac{\varepsilon}{2^n},\ x_n+\frac{\varepsilon}{2^n}\right),\ \ldots; \tag{9} \]

on each of these intervals $l_i$, constructed about the point $x_i$, there may lie other rational points, so that the intervals will overlap one another; on them there are also situated some irrational points; the least covering¹ of the intervals (9) cannot be greater than the sum of these intervals

\[ \underset{1}{\overset{\infty}{\mathrm{M}}}\left\{\frac{2\varepsilon}{2^n}\right\} < \sum_{1}^{\infty}\frac{2\varepsilon}{2^n} = 2\varepsilon. \]

The intervals $l_i$ may—first—overlap one another; from such intervals there will be formed elongated intervals of the type $A_0B_0$,² with the sole difference that the midpoints of the elongating intervals will be internal points of the elongated ones; secondly—it is possible that some of the $l_i$ are situated wholly upon others.

It is further evident that the intervals (9) cannot be adjacent; indeed, then for some $m$ and $n$ one would have to have

\[ x_m+\frac{\varepsilon}{2^m} = x_n-\frac{\varepsilon}{2^n}, \]

whence it would follow that

\[ x_n - x_m = \varepsilon\left(\frac{1}{2^m} + \frac{1}{2^n}\right), \]

i.e. the difference of two rational numbers would be irrational.

From the circumstance that the sum of all the distinct intervals cannot exceed $2\varepsilon$, it follows—on the basis of Borel’s theorem—that within the intervals (9) there are points of the interval $(0,1)$.

The least common multiple of all possible intervals (9) will constitute a countable series of intervals scattered densely over $(0,1)$, which determine a certain domain consisting exclusively of irrational points. Scheeffer was the first to point out the possibility of constructing such a domain.¹)

If a domain consists exclusively of irrational points, then rational points must be only interior points of the free intervals.

Such isolation of the rational points seems, as it were, to be at variance with the fact that every irrational number is defined as the limit of a series of rational numbers. But this is not so.

Every irrational point $\xi$ lying within an interval (9) is the limit of the endpoints of a certain series of infinitely decreasing intervals

\[ l_{m_1},\ l_{m_2},\ l_{m_3},\ \ldots,\ l_{m_\nu},\ \ldots\ ; \]

if, in these intervals, we take interior rational points, then $\xi$ will likewise be the limit of these points, owing to the fact that the intervals $l_{m_\nu}$, as they approach $\xi$, decrease indefinitely.—

Let us also note that Borel gave²) an excellent construction of free intervals for a perfect domain.

32. Of extreme interest is Borel’s definition³) of the measure of a domain, which at first glance does not coincide with the definitions that had been given before him. The course of the author’s reasoning is as follows.

Let, first, a domain consist of all points of a countable series of intervals not overlapping one another, whose sum is equal to $s$; Borel then says that “the domain has measure $s$.”

If two domains of this kind have no common points, and $s_1$ and $s_2$ are their measures, then $s_1 + s_2$ will be the measure of their sum. In general, for a countable

¹) Cf. 18°.

²) Leçons, p. 49.

³) Leçons, p. 46.

of a sequence of domains $E_i$ without common points and having measures $s_i$,

\[ s_1+s_2+\cdots+s_n+\cdots=s \]

will be the measure of their sum $E$; here it is immaterial whether or not the boundary points of the intervals defining them are included among the domains.

If $E$ contains the points $E_1$, and $s, s_1$ are their measures, then $E-E_1$ has measure $s-s_1$.

Domains for which, by virtue of the preceding conditions, a measure can be defined, Borel calls measurable (mesurable), and speaks only of measurable domains; by this name the author does not mean to assert that it is impossible to give a definition of measure for other domains, but rather that he is interested only in domains measured in the indicated sense. Borel’s definition can be expressed a little differently, and then its simplicity and naturalness are striking.

Let us first note that, both in Borel and in our exposition, by a countable sequence one may also understand a finite one; thus the domain of points $\left(\frac{1}{3},\frac{2}{3}\right)$ will have measure $\frac{1}{3}$, although the defining interval here is unique.

If the domain $E$ consists of a sequence of continuous intervals, then the sum of their lengths $s$, equal to or less than $l$—the length of the fundamental segment—is naturally taken as the measure of the domain; in this case, consequently, the measure is nothing other than length, but composed of many summands. Such a definition of measure is quite simple and natural.

If there are two domains $E_1$ and $E_2$ of this kind, but without common points, then the continuous intervals will be distributed along the fundamental segment in a certain order; they cannot overlap one another and can at most be adjacent to one another; in this latter case the common boundary can belong to only one of them, so that at least one of the intervals turns out to be open. Two countable sequences of intervals defining $E_1$ and $E_2$ again give a countable sequence, and the sum $E_1+E_2=E$ is defined by this countable sequence of intervals; it is evident that here

\[ \mathrm{M}(E_1+E_2)=s+s_1=\mathrm{M}(E_1)+\mathrm{M}(E_2). \tag{10} \]

This theorem is an immediate consequence of the definition.

If we have a sequence of domains

\[ E_1,\ E_2,\ E_3,\ldots,\ E_n,\ldots \tag{11} \]

without common points, and each of them is given by a countable sequence

intervals, then these intervals may only be adjacent, but in no way must they overlap one another.

A countable sequence of countable sequences of such intervals again gives a countable sequence of intervals, all of whose points belong to the set

\[ E = \sum E_i, \]

which is the sum of the sets (12). Since in this case

\[ \tag{12} s = \sum s_i,\quad \text{then } M \sum E_i = \sum M(E_i). \]

If, further, $E_1$ is contained in $E$, i.e. $E_1$ is given by a countable sequence of solid intervals situated in the intervals $E$, then removing $E_1$ from $E$ will have the consequence that each interval $E$ is, generally speaking, split into a countable sequence of partial intervals and that a sequence of these latter is removed.

After such a removal, the set $E_1 - E_2$ will consist of the remaining intervals, if there are any, and of points not forming solid intervals. If $E_1 - E_2$ consists only of solid intervals, whose total length is $s_1 - s_2$, the equality

\[ \tag{13} M(E_1 - E_2) = s_1 - s_2 = M(E_1) - M(E_2) \]

is again a consequence of the definition.

If, however, the set $E_1 - E_2$ contains points not constituting intervals, the difference $s_1 - s_2$ is taken as the definition of the measure; only here Borel goes beyond elementary considerations and puts forward a proposition whose inevitability can hardly be disputed, provided only that we wish to generalize the concept of length so as to extend it to sets not consisting of solid intervals.

The equalities (10) and (13) evidently retain their force also for sets that are sums of a countable sequence of other sets without common points, since each sum falls under the basic definition of measure.

Having defined the measure $s$ of a set $E$, we find—on the basis of (13)—

\[ M(l - E) = l - s. \]

From this follows the calculation of measure for closed and perfect sets, so that these too prove to be measurable in Borel’s sense.

If the set $E$ is countable, then each of its points may be regarded as a constituent set $E_i$, determined by a single interval whose length is equal to $0$; since in this case $M(E_i)=0$, then—on the basis of (12)—

\[ M(E)=M\sum_{1}^{\infty} E_i=0, \]

i.e. the measure of a countable set is always equal to zero; but not conversely: if the measure is equal to zero, this still does not mean that the set is countable. Here Borel refers to example $30^\circ$. From this it further follows that, if the measure of $E$ is not equal to zero, the set cannot be countable. Thus

Borel calls measurable all those sets which themselves or whose complementary sets can be given by a countable sequence of continuous intervals.

If every set can be given in this way, Borel’s definition of measure becomes all-embracing.

Schoenflies, speaking in his report about Borel, does not at all clarify the course of his ideas and attributes to him, among other things, something which Borel did not say. Schoenflies writes that Borel “imagines each point of the set surrounded by an arbitrary neighborhood and has to do with the part of space filled by them and with their limit.” We have seen above that Borel said nothing of the kind, and that all this runs counter to Borel’s very idea.

In exactly the same way, concerning theorem (10), Schoenflies attributes to Borel something not stated by the latter: Borel assumes that those sets for which measure may be discussed, or their complementary sets, are given by a countable sequence of continuous intervals; but Schoenflies says: “in the case where the continuum $C$ is divided in some way into two sets $E_1$ and $E_2$, it must be

\[ M(C)=M(E_1)+M(E_2); \]

this ‘in some way’ contradicts Borel’s idea”¹. Therefore Schoenflies’s objection to Borel appears to be built entirely on a misunderstanding.

33. From 1896 onward begin² the works of Schoenflies, who is chiefly interested in two-dimensional sets and—in particular—in the division of the plane by means of a closed curve.

Schoenflies is a great proponent of the arithmetical construction of a set by means of numbers written in various systems of notation, a construction often used by Peano¹.

He represents—for example—proper fractions as written in the binary system

\[ x = 0, a_1 a_2 a_3 \ldots a_n \ldots \qquad a_i = 0 \text{ or } 1, \]

excluding finite fractions in doing so, and then reads them in the decimal system. Then the set $\{x\}$ will be a nowhere dense perfect set, whose complementary intervals will be

\[ [0.(1),\, 1.0], \quad [0.0(1),\, 0.1], \quad [0.00(1),\, 0.01], \quad [0.10(1),\, 0.11], \]

\[ [0.000(1),\, 0.001], \quad [0.010(1),\, 0.011], \quad [0.100(1),\, 0.101], \quad [0.110(1),\, 0.111], \]

and so on.

The sum $S$ of these intervals proves to be

\[ S=\frac{8}{9}+\frac{8}{90}+2\cdot\frac{8}{900}+2^2\cdot\frac{8}{9000}+\ldots+2^n\cdot\frac{8}{9\cdot 10^{n+1}}+\ldots=1. \]

34. Osgood, in 1896, in an article² devoted to the question of non-uniform convergence and the integration of series, gives an interesting example and several important theorems concerning the theory of sets.

He constructs a perfect set in the following manner, calling it a particular case of Harnack’s set³:

[Diagram of a construction on the segment from $0$ to $1$, with subintervals labeled $l_1,l_2,l_3,l_4$ and points labeled $a_i^j$.]

\[ l_1 = \lambda - \frac{\lambda}{3}, \]

\[ l_1 + 2l_2 = \lambda - \frac{\lambda}{4}, \]

\[ l_1 + 2l_2 + 2^2 l_3 = \lambda - \frac{\lambda}{5}, \]

\[ \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot \]

\[ l_1 + 2l_2 + 2^2 l_3 + \ldots + 2^{n-1}l_n = \lambda - \frac{\lambda}{n+2}. \]

where $\lambda$ is an arbitrary$^4$ positive number $0<\lambda\leq 1$; whence

\[ \sum_{0}^{\infty} 2^i l_{i+1}=\lambda . \]

Then Osgood considers a countable sequence of regions

\[ \tag{1} G_1,\ G_2,\ G_3,\ldots,\ G_n,\ldots \]

such that 1) all $G_i$ are closed and nowhere dense, and 2) $G_i$ is contained in $G_{i+1}$, and defines the region

\[ \tag{2} Q=\lim_{i\ldots\infty} G_i, \]

each point of which belongs to one of the $G_i$ for sufficiently large $i$.

He then considers the region complementary to $Q$ with respect to the fundamental interval $l$, and proves that this latter region is non-denumerable in an arbitrary part of $l$. Hence it follows that

A. “The region $Q$ does not form a continuum in any part of the interval.”

Suppose further that a sequence is given

\[ \eta_1>\eta_2>\eta_3>\ldots,\quad \lim_{n\ldots\infty}\eta_n=0. \]

For a closed and nowhere dense region $G$, take those free intervals

\[ l_{11}, l_{12},\ldots, l_{1k_1},\quad \text{whose lengths are }>\eta_1, \]

\[ l_{21}, l_{22},\ldots, l_{2k_2},\quad \text{ ” \quad ” }>\eta_2, \]

and so on; let us denote

\[ \sum_{1}^{k_1} l_{1i}=\lambda_1,\quad \sum_{1}^{k_2} l_{2i}=\lambda_2,\ldots,\quad \sum_{1}^{k_n} l_{ni}=\lambda_n,\ldots; \]

then

\[ \sum_{1}^{\infty}\lambda_i=\lambda\leq l. \]

$^4$ Here $\lambda=\dfrac{3}{4}$, $l_1=\dfrac{1}{2}$, $l_2=\dfrac{1}{32}$, $l_3=\dfrac{3}{320}$, etc.

Having derived for the domain $G$ such a number $\lambda$, Osgood then defines the measure (content) of the domain by

\[ J = l - \lambda . \]

It follows from this that the points of $G$ may be included in a finite number of intervals whose sum of lengths exceeds $l-\lambda$ by less than an arbitrarily small quantity $\varepsilon$.

Finally, Osgood proves the theorem, for the domains (1) and (2),

\[ J(Q)=\lim_{i..\infty} J(G_i). \]

In 1900¹ Osgood, returning to theorem A, proves it without assuming the $G_i$ to be closed and while considering a plane domain, referring in the proof to theorem A; but even without it Osgood gives a proof of the theorem for a linear domain, using Baire’s process.²

35. In 1897 Burkhardt established³ for two-dimensional domains certain new definitions; among them, for linear domains, only the following can have significance:

He calls a domain flächenartig (surface-like) if its composition contains at least one point some neighborhood of which belongs entirely to the domain; analogously to this definition, under the same condition one may say that a linear domain is linienartig (line-like).

In general, line-like domains may be defined as those for which interior points exist. ♦

Among the line-like domains belong first of all those domains that are immediately measurable in Borel’s sense; rarely scattered perfect and countable domains do not belong to this class.

36. In one of Pringsheim’s interesting papers on the theory of the double integral⁴ we find an example of an arithmetical construction of a two-dimensional domain. Let $x$ be a proper rational fraction; let $n_x$ be the number of its decimal or $b$-ary digits, if $x$ is written in the system with base $b$. Let, further, to each rational fraction $x$ there also correspond fractional rational values $y$, for which the number of digits $n_y$ is equal to $n_x$.

Then for every rational $x$ there exists a finite number of values $y$, i.e., on each vertical line with rational abscissa there lies a finite number of points of the domain $P$. Obviously, the same is true also with respect to each rational horizontal line¹.

Let us now see how the points of the domain are situated on the straight line

\[ y = a + x, \tag{1} \]

where $a$ is also a rational fraction.

From equation (1), for every $x$ with $n_x > n_a$ it is possible to determine one point of the domain lying on the straight line; the value $x$ for which $n_x = n_a$ can determine the corresponding value $y$ only in the case when the final digits of $x$ and $a$ do not complete one another to $0$ or to $b$; finally, if $n_x < n_a$, the abscissa $x$ does not correspond in (1) to any point $P$.

Thus, to every rational $x$ with a number of digits greater than $n_a$ there corresponds on the straight line (1) a point $P$; consequently, the points $P$ will be densely scattered along the straight line (1).

Assigning to $a$ positive and negative fractional values, we obtain a densely scattered domain of parallel straight lines. Hence it follows that the points $P$ will be densely scattered over the square $(0,1)$, although on each rational horizontal or vertical line only a finite number of them can lie, while on irrational straight lines there are no points of the domain at all.

This example is of great interest in the theory of double and multiple integrals; it is no less important in the theory of domains, since it shows that, having studied linear domains in detail, we are still very far from being able to draw conclusions from this concerning the structure of two-dimensional domains; for the latter a completely new investigation is required, which was begun by the works of Schoenflies, and which must be based on certain new propositions.

37. In 1899 Baire, in his memoir concerning functions of a real variable², made extensive use of the theory

domains, and introduces some new results into this theory. In doing so, speaking of Cantor’s transfinite numbers, Baire notes¹ that he will not concern himself with the difficulties that are connected with the abstract notion of a transfinite number, although he uses them as a convenient language for denoting quite definite phenomena.

Passing to the construction of a rarely scattered perfect domain by means of a countable sequence of free intervals $\{l_i\}$, Baire says: arrange the intervals in some order, for example—by magnitude. Removing from the basic interval $l$ the intervals

\[ l_1,\ l_2,\ l_3,\ldots l_n, \]

we obtain $\{\lambda_j\}$—a finite sequence of the remaining intervals, on which the intervals

\[ \tag{1} l_{n+1},\quad l_{n+2},\ldots ; \]

are situated; in this way all points of the perfect domain will lie on $\{\lambda_j\}$, and the limits of the endpoints of the intervals $\{l_i\}$ that do not themselves serve as endpoints will turn out to be interior points of $\{\lambda_j\}$.

Let $\lambda^{(n)}$ be the largest of the intervals $\{\lambda_j\}$; with an infinite continuation of the process, from the intervals $\{\lambda_j\}$ the intervals (1) are successively removed; hence the $\lambda_j$ decrease. It is not difficult to see that it must be

\[ \lim_{n\ldots\infty}\lambda^{(n)}=0. \]

Indeed: if this were not so, and it were—therefore—

\[ \lim \lambda^{(n)}=\lambda\ne 0, \]

then from the interval $\lambda$ we could not remove the free intervals (1); then all points of $\lambda$ would be included in $P$, which is impossible, since $P$—by hypothesis—is rarely scattered.

Usually the theory of domains deals with the distribution of points over the continuum; Baire takes an arbitrary perfect domain $P$ and considers² a point domain $G$ with respect to the domain $P$; this reduces, in essence, to the fact that the points of the domain $G$ belong to $P$.

In particular, assuming for a rarely scattered and closed set $P$ the existence of the derivative $P^{(\Omega)}$, he takes the closed set $P_1$ with respect to $P^{(\Omega)}$; then, assuming that $P_1$ is of the second kind, so that $P_1^{(\Omega)}=0$, he takes the closed set with respect to $P_1^{(\Omega)}$ and calls it $P_2$. Continuing this process infinitely, analogous to Cantor’s process, he obtains $P_\omega, P_{2\omega}$, etc., and finally the set $P_\Omega$, which, analogously with $P^{(\Omega)}$, necessarily turns out to be perfect. Concerning the appearance of $P_\Omega$, Baire argues$^1$ as follows: for every point of the set $P$ either there exists a subsequent $P_x$ into which it enters, or there is no such $P_x$; in the latter case the points possessing this property constitute $P_\Omega$. These sets play an essential role in Baire’s investigation.

Finally Baire establishes$^2$ an extremely important and new concept:

Let there be a countable sequence of rarely scattered sets

\[ P_1,\ P_2,\ P_3,\ldots,\ P_n,\ldots; \tag{2} \]

take the set of the various points entering into (2),

\[ P = M\{P_1, P_2, P_3,\ldots, P_n,\ldots\}; \tag{3} \]

it is easy to see that in every part of the interval there exist points not belonging to $P$. Indeed—in the derived part of the fundamental interval $l$ one may take an interval $(\alpha_1,\beta_1)$ free of points of $P_1$, within it an interval $(\alpha_2,\beta_2)$ free of points of $P_2$, and so on; in this way we obtain

\[ \lim a_i = \alpha \leq \beta = \lim \beta_i . \]

Whether $\alpha=\beta$ or $\alpha<\beta$, in any case there is at least one point which certainly does not enter into any $P_i$ and—consequently—does not belong to the composition of $P$.

Sets of the form (3) Baire calls sets of the first category, and all sets not possessing this structure—sets of the second category.

The set $P$ may be of a nature quite different from the $i$; it may be—for example—everywhere dense on the interval and not closed, while the sets $P_i$ are rarely scattered and closed; etc.$^3$

$^1$ P. 51.
$^2$ P. 65.
$^3$ See Chapter II.

Concerning the nature of the set $P$ and of its complementary one, Schoenflies¹ says that it is of “fundamental importance”; these sets serve as the basis for conclusions in many theorems of analysis and therefore require the closest investigation². Schoenflies constructs $P$ also under the condition that the sum of the intervals occupied by the points of each of the $P_i$ is arbitrarily small³.

Since the sets $P_i$ cannot fill the whole interval $l$, the continuum belongs to the number of sets of the second category.

The sum of a finite or countable number of sets of the first category likewise gives a set of the first category. Indeed: let there be given rows of nowhere dense sets

\[ \begin{aligned} & P_{11},\ P_{12},\ P_{13},\ldots,\ P_{1n},\ldots,\quad P_1 \equiv \mathrm{M}\{P_{1i}\},\\ & P_{21},\ P_{22},\ P_{23},\ldots,\ P_{2n},\ldots,\quad P_2 = \mathrm{M}\{P_{2i}\},\\ & \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\\ & P_{n1},\ P_{n2},\ P_{n3},\ldots,\ P_{nn},\ldots,\quad P_n \equiv \mathrm{M}\{P_{ni}\},\\ & \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ ; \end{aligned} \]

call

\[ P \equiv \mathrm{M}(P_1,\ P_2,\ P_3,\ldots) \equiv \mathrm{M}\{P_{ij}\}. \]

The double row of nowhere dense sets $P_{ij}$ can easily be transformed into a simple row, if we put—for example—

\[ Q_1 \equiv P_{11},\quad Q_2 \equiv P_{12}+P_{21},\quad Q_3 \equiv P_{13}+P_{22}+P_{31},\ldots,\quad Q_n \equiv \sum_{i=1}^{n} P_{i(n+1-i)}; \]

then we shall have

\[ P \equiv \mathrm{M}\{Q_1,\ Q_2,\ Q_3,\ldots Q_n,\ldots\}. \]

Since each of the sets $Q_i$ consists of a finite number of nowhere dense sets, $P$—on the basis of Baire’s definition—is a set of the first category.

The set $\Pi$, complementary to a set of the first category, will be a set of the second category. Indeed: suppose that this is not true.

Then $\Pi$ will be a domain of the first category and—consequently—can be obtained as

\[ \mathop{\mathrm M}_{1}^{\infty}\{\Pi_i\}, \]

where

\[ \mathop{\mathrm M}_{1}^{\infty}\{P_i\}+\mathop{\mathrm M}_{1}^{\infty}\{\Pi_i\} = \mathop{\mathrm M}_{1}^{\infty}\{P_i,\Pi_i\} = \{l\}, \]

i.e. in this case the domain $\mathop{\mathrm M}_{1}^{\infty}\{P_i,\Pi_i\}$ would have to give all the points of $l$, which is contrary to the definition of a domain of the first category.

If $P$ is a domain of the first category, $\Pi$ its complement, and $K$ some other domain of the second category, then $\Pi$ and $K$ necessarily have common points.

Indeed: if $D(K,\Pi)=0$, then $K=D(P)$, which cannot be, since $K$ is of the second category.

“From this follows a profound difference between domains of the two categories, based neither on size nor on density in an interval, since domains of the first category may have the size of the continuum and may also be everywhere dense, but rather—on a certain combination of both these notions”¹.

Baire, having in view the interests of the theory of functions, still considers it necessary² to create a more general theory, comprising, as a special case, the theory of point domains and $n$ dimensions; these ideas of the author are far removed from the substance of the present work.

38. In an article³ devoted to the theory of functions, de-Stefano in 1900 gives the following construction of a point domain:

Divide the interval $l$ into $n_1$ equal parts

\[ l_1,\ l_2,\ l_3,\ldots,\ l_{n_1} \]

and, removing one of them $l_\nu$, the author divides, from the remaining $n_1-1$, each $l_i$ into $n_2$ equal parts

\[ l_{i1},\ l_{i2},\ l_{i3}\ldots,\ l_{in_2}; \]

again deleting in each $l_i$ one of the intervals $l_{i\nu_i}$, de-Stefano assumes the process to continue to infinity. The domain of division points obtained thereby will be nowhere dense.

Indeed: take on $l$ some interval $(\alpha,\beta)$, and suppose moreover that

\[ \beta-\alpha \geq \frac{l}{n_1 n_2 \ldots n_{m-2}} ; \]

then in $(\alpha,\beta)$ there lies at least one of the $\dfrac{1}{n_1 n_2 n_3 \ldots n_{m-1}}$-th parts of the whole interval $l$, and in it a free interval equal to

\[ \frac{l}{n_1 n_2 n_3 \ldots n_{m-1} n_m}. \]

The free intervals after the first, second, $\ldots$, $n$-th division will constitute

\[ \frac{1}{n_1},\quad \frac{n_1-1}{n_1 n_2},\quad \frac{(n_1-1)(n_2-1)}{n_1 n_2 n_3},\quad \ldots,\quad \frac{(n_1-1)(n_2-1)\ldots(n_{m-1}-1)}{n_1 n_2 \ldots n_{m-1} n_m} \]

part of $l$, and their sum

\[ \delta = \left\{ \frac{1}{n_1} + \frac{n_1-1}{n_1}\cdot\frac{1}{n_2} + \frac{n_1-1}{n_1}\cdot\frac{n_2-1}{n_2}\cdot\frac{1}{n_3} +\ldots+ \frac{n_1-1}{n_1}\cdot\frac{n_2-1}{n_2}\ldots \frac{n_{m-1}-1}{n_{m-1}}\cdot\frac{1}{n_m} +\ldots \right\}l; \tag{1} \]

the sum of the intervals on which the points of the set are situated will then be

\[ s=l-\delta = \frac{n_1-1}{n_1}\cdot \frac{n_2-1}{n_2}\ldots \frac{n_{m-1}-1}{n_{m-1}}\ldots l. \tag{2} \]

Depending on the choice of the numbers $n_i$, it may be that

\[ \frac{n_1-1}{n_1}\cdot \frac{n_2-1}{n_2}\ldots \frac{n_{m-1}-1}{n_{m-1}}\ldots = 0 \quad\text{or}\quad >0; \]

the first occurs—for example—if all $n_i$ are equal to one another, the second—if

\[ n_1=n>1,\quad n_2=n^2,\ldots,\quad n_m=n^m,\ldots ; \tag{3} \]

the first is self-evident, while the second follows from the fact that—by virtue of (1) and under condition (3)—

\[ \delta< \left\{ \frac{1}{n^1} + \frac{1}{n^2} + \frac{1}{n^3} +\ldots+ \frac{1}{n^m} +\ldots \right\}l = \frac{l}{n-1}. \]

Besides the construction of the preceding set, de-Stefano gives another theorem:

“If the set $P$ is closed, and for it $s>0$, then the part of it consisting of two-sided limiting points is uncountable.”

39. In 1900 there appeared¹ an extensive survey, belonging to Schoenflies, of the theory of domains and its applications; in it the author set himself the aim—so to speak—of popularizing this theory and attracting to it the wider attention of the mathematical world; in view of this, the first two chapters² have, by the author’s own admission, the character of a textbook. In this work the author acknowledges as his own certain results and methods of proof and, what is most important, the genetic development that he gave to his survey³.

Of all this, we consider it necessary to note here what has added to our stock of information concerning the theory of domains, and what illuminates already known results from a new point of view.

Let us note⁴ first of all the theorem converse to Borel’s theorem:

“Let $P = \{x\}$ be a domain that is everywhere dense on the fundamental interval $l$, and suppose that each point $P$ is interior to one of the intervals of the domain $\{l_i\}$; then the non-interior points of the intervals $\{l_i\}$, if they exist, form a nowhere dense closed domain $Q$,” because a limiting point $Q$ cannot lie inside $\{l_i\}$.

The same theorem applies, to a greater extent, also to a nowhere dense domain $P$.

Let us note further a new formulation of the Cantor–Bendixson theorem⁵:

“A countable domain can be exhausted by a countable sequence of isolated domains; an uncountable domain, after the separation of a countable sequence of isolated domains, is reduced to a perfect domain.”

As an example of the gradual exhaustion of a countable domain, Schoenflies singles out⁶ from the domain of rational fractions the domain $P_1$ of those fractions whose denominators are powers of $2$; then the domain $P_2$ with denominators of the type $3^n$; and so on; after singling out the countable sequence of domains $P_1, P_2, P_3, \ldots$, we obtain the domain of fractions whose denominators are composite numbers; from these we successively single out the domains of fractions with denominators of the form

\[ (2.3)^n,\qquad (2.5)^n,\qquad (2.7)^n,\ldots; \]

then with denominators of the form

\[ (3.5)^n,\qquad (3.7)^n,\qquad (3.11)^n,\ldots; \]

and so on.

A closed domain $P$ is given by a countable sequence $D=\{\delta_i\}$ of intervals $\delta_i$ that do not overlap one another.

If $P_1$ is a closed domain forming part of $P$, then its domain of intervals $D_1=\{\delta'_i\}$ will be obtained from $D$ either by shortening the entire fundamental interval, or by joining adjacent or non-adjacent intervals $\delta_i$ into one; generally speaking, the passage from $D$ to $D_1$ will be accompanied by the disappearance of a countable sequence of intervals. If $P_2$ is likewise a closed component of $P_1$, and $D_2$ is the corresponding domain of intervals, then $D_2$ is derived from $D_1$, again by decreasing the number of free intervals, and so on. Since the countable sequence of intervals $D$ will be exhausted by a countable sequence of such operations, we may say$^1$ that

“a closed domain can be decomposed into a countable sequence of closed domains.”

Schoenflies’s attempt to construct free intervals for a two-dimensional domain$^2$ is connected with his later works concerning analysis of position, and they go beyond the limits of the present investigation; it is only necessary to note that, in our view, greater results in the interests of the theory of domains may be expected along the path entered upon by Zoretti.

In asking$^3$ about the necessary and sufficient conditions under which the measure of a perfect domain is equal to zero, Schoenflies constructs such a domain $P$ in the following way:

Let $l$ be the fundamental interval, and let $\{l_i\}$ be the intervals free of points of $P$, arranged by magnitude; call

\[ l=t_1,\qquad t_1-l_1=t_2,\qquad t_2-l_2=t_3,\ldots,\qquad t_{n-1}-l_{n-1}=t_n,\ldots, \]

where $t_n$ is the sum of the intervals occupied by the points of $P$ and remaining after the removal of the first $n-1$ free intervals. If we further call $\varepsilon_n$ the ratio $l_n:t_n$, whence $l_n=\varepsilon_n t_n$, we shall have

\[ t_1=l,\qquad t_2=(1-\varepsilon_1)t_1,\qquad t_3=(1-\varepsilon_2)t_2,\ldots,\qquad t_n=(1-\varepsilon_{n-1})t_{n-1},\ldots, \]

whence

\[ t_n=(1-\varepsilon_1)(1-\varepsilon_2)\ldots(1-\varepsilon_{n-1})\,l, \]

and the measure of the domain $P$

\[ t=\lim_{n\ldots\infty} t_n = l.\lim \prod_1^n(1-\varepsilon_i). \]

$^1$ ib. S. 80.
$^2$ ib. S. 81–85.
$^3$ ib. S. 94.

For $t$ to be equal to zero, it must be that

\[ \prod_{1}^{\infty}(1-\varepsilon_i)=0; \]

this is precisely—according to Schoenflies—the desired condition.

Giving further a modification of a similar construction, where enter the ratios of each $l_n$ to the intervals adjacent to it that are occupied by the points $P$ and enter into the composition of $t_n$, the author thinks¹ that further work in this direction is desirable. Some results in the second chapter of the present investigation are connected with these considerations.

To Baire’s theorems concerning domains of the second category Schoenflies adds one more:

“Common points of two domains $P$ and $K$ of the second category also give a domain of the second category.”

Indeed: let

\[ P=\mathop{\mathrm M}_{1}^{\infty}\{P_i\}, \qquad Q=\mathop{\mathrm M}_{1}^{\infty}\{Q_i\}, \]

be domains of the first category, complementary to the domains $P$ and $K$; then for the domains

\[ \mathrm M(P_1,Q_1), \qquad \mathrm M(P_2,Q_2), \ldots, \qquad \mathrm M(P_n,Q_n), \ldots \]

the domain

\[ \mathop{\mathrm M}_{1}^{\infty}\{P_i,Q_i\}=L \]

will prove to be of the first category, and its complement $\Lambda$—of the second category; while the domain $\Lambda$ consists of the common points of $P$ and $K$.

With regard to the domains $P$ and $П$, Schoenflies makes one more remark: neither the one nor the other will necessarily be closed, and both may be uncountable; the author poses the question² whether every everywhere dense, non-closed and uncountable domain will always be a domain of the first or of the second category?

Speaking of Baire’s domains, referred to continuous space and to perfect domains, and of the possibility of establishing, with respect to them, theorems analogous to the theorems concerning the continuum, Schoenflies sees in this³ a certain equivalence

of all perfect sets and to find that here lies one of the most important results of the theory of sets.

40. The development of Borel’s ideas concerning the measure of a set was to be served by the investigations of Lebesgue, who says¹ that he “completed and made more precise Borel’s somewhat hasty indications”; we shall see below that this statement does not entirely accord with reality. Lebesgue approaches the concept of measure in the following way:

“With each bounded set we set ourselves the aim of associating some positive number or zero, which we call its measure, and we subject it to the following conditions:

1) There exist sets whose measure is not equal to zero;

2) Equal sets have equal measures;

3) The measure of the sum of a finite or countable sequence of sets having no common points is equal to the sum of their measures”;

later,² in 1904, instead of 1) another condition was introduced:

“1) The measure of the set of all points of the interval $(0,1)$ is equal to unity.”

The sets for which the measure determined in this way exists are called measurable by Lebesgue. If a definition satisfying these conditions is possible,³ a set consisting of one point will have measure $0$, since a finite set consisting of an infinite number of points must have a finite measure; the measure of an open or closed segment cannot be equal to zero, since otherwise the same would be true of every finite segment; as the unit one may take the measure of the set of points of any segment, namely—its length.

Having set out these remarks, Lebesgue passes to the very definition of measure, diverging—in our view—from Borel in the very idea.

Assuming the set $\mathrm{P}$ to be arbitrary, he encloses it in intervals $\{l_i\}$, denoting by $\mathrm{P}_1=\{l_i\}$ the set of all points situated on them.

Since the points of $\mathrm{P}$ enter wholly into the composition of $\mathrm{P}_1$, we have

\[ \tag{1} \mathrm{m}(\mathrm{P}) < \mathrm{m}(\mathrm{P}_1)=\sum_1^{\infty} l_i; \]

for all possible choices of $l_\nu$,

\[ \mathrm{m}(P)\leq \operatorname{uGr}\sum_{1}^{\infty} l_\nu; \]

Lebesgue calls this lower bound the outer measure of the domain $P$:

\[ \mathrm{m}_e(P)=\operatorname{uGr}\sum_{1}^{\infty} l_\nu, \]

so that

\[ \mathrm{m}(P)\leq \mathrm{m}_e(P). \tag{2} \]

Defining the outer measure also for the domain $\Pi$—the complement of $P$ with respect to the fundamental segment $l$—we have

\[ \mathrm{m}(\Pi)\leq \mathrm{m}_e(\Pi); \tag{3} \]

since—by virtue of condition 3)—it must be$^1$

\[ \mathrm{M}(P)+\mathrm{M}(\Pi)=l, \tag{4} \]

then, by (3),

\[ \mathrm{m}(P)\geq l-\mathrm{m}_e(\Pi); \tag{5} \]

this number

\[ l-\mathrm{m}_e(\Pi)=\mathrm{m}_i(P) \tag{6} \]

Lebesgue calls the inner measure of $P$.

From (6) and (5) it follows that

\[ \mathrm{m}_i(P)+\mathrm{m}_e(\Pi)=l, \tag{7} \]

\[ \mathrm{m}(P)\geq \mathrm{m}_i(P); \tag{8} \]

by virtue of (2) and (8),

\[ \mathrm{m}_e(P)\geq \mathrm{m}(P)\geq \mathrm{m}_i(P), \tag{9} \]

if only the problem of measurement is possible in Lebesgue’s sense.

It must be observed that the derivation of inequalities (9) cannot be regarded as irreproachable, since it is based on inequality (1), which—in our view—is not sufficiently motivated.

Domains for which

\[ \mathrm{m}_e(P)=\mathrm{m}_i(P). \]

$^1$ Leçons, p. 104.

the author calls measurable; this definition—by virtue of (4)—is equivalent to the following:

“An area $P$ is measurable if it is possible to enclose it and its complement $\Pi$ in such intervals that the sum of their common parts will be arbitrarily small.”

We see, therefore, that Lebesgue’s definition is a definition in the old style, and by no means coincides in idea with Borel’s definition, based on continuous intervals independent of any arbitrariness whatsoever. Moreover, here it still has to be proved that this definition of Lebesgue’s does not contradict the three conditions indicated above; as regards 1) and 2), this is self-evident; for 3), however, one must prove that the sum of a countable series of measurable areas $P_1, P_2,\ldots, P_\nu,\ldots$ without common points will also be measurable, and that the measure of the sum is equal to the sum of the measures¹.

Let

\[ \sum_1^\infty \varepsilon_\nu=\varepsilon, \]

where $\varepsilon_\nu$ are certain small positive quantities; since all $P_\nu$ are measurable, it is possible to enclose $P_\nu$ and their complementary areas $\Pi_\nu$ in such intervals with sums $\alpha_\nu$ and $\beta_\nu$, that the common part of $\alpha_\nu$ and $\beta_\nu$ will be equal to $\varepsilon_\nu$.

If we call $\alpha'_2, \beta'_2$ those parts of $\alpha_2, \beta_2$ which lie on $\beta_1$; $\alpha'_3, \beta'_3$—the parts of $\alpha_3, \beta_3$ which lie on $\beta'_2$, and so on, and put, for symmetry, $\alpha'_1=\alpha_1$, then—obviously—the sum of the intervals enclosing the points of all $P_\nu$ will be equal to

\[ \alpha_1+\alpha'_2+\alpha'_3+\cdots+\alpha'_\nu+\cdots=s; \]

whence

\[ \mathrm{m}_e(P)=\mathrm{uGr}\,s. \]

According to the definition of measure,

\[ \sum_1^n [\mathrm{m}(P_\nu)-\varepsilon_\nu]\leq \sum_1^n M\{\alpha_\nu\}=\sum_1^n \alpha'_\nu; \]

whence

\[ \sum_1^\infty \mathrm{m}(P_\nu)\leq s+\varepsilon. \tag{11} \]

Since further

\[ \alpha_\nu-\mathrm{m}(P_\nu)\leq D(\alpha_\nu,\beta_\nu)=\varepsilon_\nu, \]

then

\[ \alpha'_\nu\leq \alpha_\nu\leq \mathrm{m}(P_\nu)+\varepsilon_\nu; \]

therefore

\[ s=\sum_{1}^{\infty}\alpha_\nu' \leq \sum_{1}^{\infty} \mathrm{m}(P_\nu)+\varepsilon; \tag{12} \]

from comparing (11) and (12) we have

\[ \sum_{1}^{\infty}\mathrm{m}(P_\nu)-\varepsilon < s \leq \sum_{1}^{\infty}\mathrm{m}(P_\nu)+\varepsilon; \tag{13} \]

whence

\[ \mathrm{m}_i(P)=\underline{\mathrm{Gr}}\,s=\lim s=\sum_{1}^{\infty}\mathrm{m}(P_\nu). \tag{14} \]

On the other hand, the domain $\Pi=\mathrm{D}_{1}^{\infty}(\Pi_\nu)$, i.e. it is contained in each of the domains $\Pi_\nu$; therefore the points of $\Pi$ can be enclosed in intervals with sum $\beta_n'=\mathrm{D}_{1}^{n}(\beta_\nu)$, as a consequence of which $\mathrm{m}_e(\Pi)\leq \beta_n'$.

But

\[ \sum_{1}^{n}\alpha_\nu' + \beta_n' < l+\sum_{1}^{n}\varepsilon_\nu, \]

therefore

\[ \mathrm{m}_e(\Pi)\leq l-\sum_{1}^{n}\alpha_\nu' + \sum_{1}^{n}\varepsilon_\nu = l-s+\sum_{1}^{n}\varepsilon_\nu+\sum_{n+1}^{\infty}\alpha_\nu'; \]

since the series $\sum_{1}^{\infty}\alpha_\nu'$ is convergent, the number $n$ can be chosen so that

\[ \sum_{n+1}^{\infty}\alpha_\nu' < \varepsilon; \]

then—in virtue of (13)—

\[ \mathrm{m}_e(\Pi)\leq l-s+2\varepsilon \leq l-\sum_{1}^{\infty}\mathrm{m}(P_\nu)+\varepsilon; \]

whence it follows that

\[ \mathrm{m}_i(P)=l-\mathrm{m}_e(\Pi)\geq \sum_{1}^{\infty}\mathrm{m}(P_\nu)-\varepsilon, \]

and, in connection with (9) and (14),

\[ \sum_1^\infty m(P_\nu) - \varepsilon < \overline{m_i}(P) \leq \overline{m_e}(P) = \sum_1^\infty m(P_\nu), \]

whence it follows that

\[ m_i(P) = m_e(P) = \sum_1^\infty m(P_\nu), \]

as a result of which the domain $P$ turns out to be measurable, and

\[ m(P) = \sum_1^\infty m(P_\nu). \]

Thus, admitting the definition of Lebesgue measure, we see that it satisfies the conditions imposed upon it.

If the domains $P_\nu$ had common points, then one could likewise prove that

\[ \overset{\infty}{M}_1\{P_\nu\} \]

will be measurable, but only

\[ m(P) < \sum_1^\infty m(P_\nu). \]

Besides finding the measure of a sum, it is often necessary to find the measure of the domain $P$ of points which belong to all domains of a countable sequence of domains $\{P_i\}$.

Having taken a sequence of supplementary domains $\{\Pi_\nu\}$, we see that the common points of all the $P_\nu$ do not belong to any one of the $\Pi_\nu$; if the least multiple of all the $\Pi_\nu$ is denoted by

\[ \overset{\infty}{M}_1\{\Pi_\nu\} = \Pi, \]

then $P$ will be the supplementary domain to $\Pi$. If the $P_\nu$, and consequently the $\Pi_\nu$, are measurable, then $\Pi$ is measurable, and also $P$.

Next, if $P_1$ contains $P_2$,

\[ P_1 - P_2 = M\{P_1,\Pi_2\}; \]

therefore, if $P_1$ and $P_2$ are measurable, then $P_1-P_2$ will also be measurable.

Thus, by carrying out upon a measurable region the operations indicated above, we shall always obtain measurable regions; but it still does not follow from this¹ that the problem of measurement is impossible for those regions for which $m_i(P) \ne m_e(P)$.

Next, Lebesgue finds it of interest to compare regions measurable in Jordan’s sense and in his own sense.

If we compare² the exterior content $E(P)$ of a region $P$ with Lebesgue’s outer measure, the following will turn out to be true: the former includes only the interior and boundary points of the region $P$; therefore, without going beyond the bounds of $E(P)$, we can choose intervals in such a way that they will contain the whole region $P$; consequently

\[ E(P) \ge m_e(P); \tag{15} \]

since, further,

\[ e(P)=l-E(\Pi), \qquad m_i(P)=l-m_e(\Pi), \]

and, similarly to (15),

\[ E(\Pi) \ge m_e(\Pi), \]

then

\[ e(P) \le m_i(P); \]

therefore

\[ E(P) \ge m_e(P) \ge m_i(P) \ge e(P). \]

It follows from this that, if a region is measurable according to Jordan, i.e. if $E(P)=e(P)$, then all the more $m_e(P)=m_i(P)$, i.e. it is also measurable according to Lebesgue, but not conversely.

In assessing the significance of Lebesgue’s method, we must acknowledge that in fact, while differing from Borel in idea, the author defines measure only for regions measurable according to Borel; the finding of $m_e(P)$ and $m_i(P)$ presents considerably greater difficulties than the determination of $E(P)$ and $e(P)$ by Jordan, which the author himself does not deny³.

Thus—in our view—Lebesgue has decidedly not advanced the question of the measure of a region in comparison with Borel; at the same time, he has neither simplified nor made more precise the readily accessible and quite simple ideas of Borel.

Lebesgue then applies his method to the plane, taking a triangle as the elementary interval.

41. In 1903 Borel, speaking of the construction of intervals of type $29^\circ$, $30^\circ$, notes the essential difference between two procedures: one—when the fundamental interval is divided into partial intervals, and the law of division is not determined by the point set which is intended; and another—when the given points are taken as the point of departure, and intervals are constructed around these points. How different these two procedures are is seen from the following example: if the segment $(0,1)$ is divided into parts, then, whatever they may be, rational points will lie on them, and the sum of the partial intervals containing such points is equal to one; but if one constructs segments of type $30^\circ$ around each rational point $\dfrac{p_i}{q_i}$, then one obtains a countable set of intervals on which such points, and only such points, are situated; moreover, the sum of these intervals can be made arbitrarily small.

In dealing with the question of the approximate representation of irrational numbers by means of rational fractions, Hurwitz in 1891¹ established the theorem: “to every irrational number $\alpha$ one can approach by means of an infinite sequence of irreducible rational fractions $\dfrac{p_i}{q_i}$ in such a way that

\[ \left|\alpha-\frac{p_i}{q_i}\right|<\frac{1}{q_i^2\sqrt{5}}. \]

Borel, following Hurwitz, calls² the interval

\[ \left(\frac{p}{q}-\frac{1}{q^2\sqrt{5}};\ \frac{p}{q}+\frac{1}{q^2\sqrt{5}}\right) \]

the canonical interval for the fraction $\dfrac{p}{q}$. Hurwitz’s theorem may be paraphrased thus: “every irrational number is contained inside an infinite set of canonical intervals.” Borel adds to this that the same will also be true for every commensurable number, if one admits that the fraction $\dfrac{p}{q}$ may be expressed, for every integer $n$, in the form $\dfrac{np}{nq}$, i.e. if reducible fractions too are admitted.

It follows from this that every number of the interval $(0,1)$ will be interior to at least one canonical interval, the number of which is infinite.

And in such a case—by Borel’s theorem—one can, in infinitely many ways, choose a finite number of such canonical intervals, so that every point of $(0,1)$ will lie in the interior of at least one of them. Borel calls a system of fractions defining such intervals a complete system.

Borel then extends his theorem as follows: “If in $n$-dimensional space there is an $E$—a bounded closed domain—and $P_1, P_2, P_3,\ldots,P_n,\ldots$ is a countable sequence of such domains that every point of $E$ is an interior point of at least one of the $P_n$, then among the $P_n$ there exists a finite number of domains with the same property.”

In proving this, the author uses his second method, based on the numbering of intervals.

42. In 1903 Lindelöf gives $^{1}$ the following formulation of Borel’s last theorem:

A. “If in $n$-dimensional space, about every point of a bounded closed domain $P$, a ball is described, then one can choose a finite number of balls so that every point of $P$ will be an interior point of at least one ball.”

A generalization of this theorem is the theorem:

B. “If the domain $P=\{x\}$ is arbitrary, and we construct balls of variable radius $\rho_x$, then it is possible to choose a countable sequence of balls with the same property.”

Indeed: a) if the domain does not extend to infinity, and all $\rho_x$ are greater than some $\rho_0$, it is clear that a finite number of balls can be chosen; b) if the domain does not extend to infinity, but the $\rho_x$ may be arbitrarily small, divide the domain $P$ into parts $P_1,P_2,P_3,\ldots,P_n,\ldots$, where $P_n$ contains all points for which

\[ \varepsilon_{n-1}\geqq \rho_x > \varepsilon_n, \]

where $\varepsilon_1,\varepsilon_2,\varepsilon_3,\ldots$ is a sequence of positive numbers decreasing without bound; then each $P_n$ satisfies the conditions of a), and in this case the whole domain $P$ can be included in a countable sequence of balls; finally c) in the general case $P$ can be divided into a countable sequence of finite parts.

If a point $x$ of the domain $P$ can be surrounded by some ball inside which there will be only a countable sequence of other points of $P$, then Lin-

$^{1}$ C. R., t. 137, p. 697.

Lindelöf calls a $P$-domain countable in the neighborhood of a given point. This notion is a development of the notion of isolated and limit points: if inside some sphere about a given point $x_0$ there is a finite number of points of $P$, it will be an isolated point; if the number of points is countable, it may be called a countable limit point; in the contrary case we shall have an uncountable limit point; these last points Lindelöf calls points of condensation. Having established these definitions, the author gives the theorem:

C. “If a domain is countable in the neighborhood of each point, then it is countable.”

From this it is easy to derive, without the aid of transfinite numbers, the Cantor–Bendixson theorem:

D. “Every uncountable closed domain $P$ is composed of a perfect and a countable part.”

Let us denote by $U$ the domain of those points of $P$ in whose neighborhood $P$ is countable, and by $S$ the domain of the remaining points,

\[ P \equiv U + S . \]

On the basis of theorem C the domain $U$ is countable; as for $S$, it is clear that each of its points is a limit point, and each of its limit points $\xi$, being a limit point of $P$, belongs to $P$, since $P$ is closed; consequently $\xi$, being an uncountable limit point, enters into $S$; thus $S$ will be perfect.

Passing to Cantor’s definition of the measure of domains $P$, if it is closed and finite, Lindelöf describes spheres about the points $x$ of the domain $P$ with radii $\rho_x$; let us call the volume of the part of space occupied by them $\Pi(\rho_x, P)$; if all radii are equal to $\rho$, then, for $\rho > \rho_x$,

\[ \tag{1} \Pi(\rho, P) \geqq \Pi(\rho_x, P); \]

according to Cantor’s definition $^{1}$

\[ \lim_{\rho \ldots 0} \Pi(\rho, P) = I(P) \]

is the measure (Inhalt) of the domain; let us prove that

E. “\[ \lim_{\rho \ldots 0} \Pi(\rho, P) = \lim_{\rho_x \ldots 0} \Pi(\rho_x, P). \]”

Indeed: having taken $\Pi\!\left(\dfrac{\rho_x}{2}, P\right)$, we can—on the basis of A—choose a finite number $\mu$ of spheres with radii

\[ \frac{\rho_1}{2},\ \frac{\rho_2}{2},\ \ldots,\ \frac{\rho_\mu}{2}, \]

inclu—

$^{1}$ See 17°.

containing within them all the points $P$; let us denote by $\Pi_{\mu}(\rho_x, P)$ the sum of the corresponding $\mu$ balls in $\Pi(\rho_x, P)$; then

\[ \Pi(\rho_x, P) \supseteq \Pi_{\mu}(\rho_x, P). \tag{2} \]

Let us also call $\alpha$ the least of the radii $\frac{\rho_1}{2}, \frac{\rho_2}{2}, \ldots, \frac{\rho_\mu}{2}$; if we describe about all the points of the region $P$ balls of radius $\alpha$, then $\Pi(\alpha, P)$ will cover all the points $P$. On the other hand, the $\mu$ balls of radii $\frac{\rho_1}{2}, \frac{\rho_2}{2}, \ldots, \frac{\rho_\mu}{2}$ likewise contain within them all the points $P$; consequently the $\mu$ balls of radii $\rho_1, \rho_2, \ldots, \rho_\mu$, i.e. $\Pi_{\mu}(\rho_x, P)$, not only cover all the points of the region $P$, but moreover do this in such a way that the distance of any point $P$ from the boundary of $\Pi_{\mu}(\rho_x, P)$ will be not less than $\alpha$. Therefore all the balls $\alpha$ lie inside the $\mu$ balls $\rho_x$, i.e.

\[ \Pi_{\mu}(\rho_x, P) \supseteq \Pi(\alpha, P); \tag{3} \]

from this it follows immediately that $\Pi(\alpha, P)$ is contained in a finite number, not exceeding $\mu$, of separate parts of space; $\Pi(\rho_x, P)$ will have the same property if all $\rho_x$ are less than $\alpha$.

On the basis of (1), (2), and (3)—with $\rho > \rho_x$—

\[ \Pi(\rho, P) \supseteq \Pi(\rho_x, P) \supseteq \Pi_{\mu}(\rho_x, P) \supseteq \Pi(\alpha, P), \tag{4} \]

whence theorem E follows. From E there further follow Cantor’s theorems:

F. “The measure of a closed countable region is equal to zero,” since the $\rho_x$ may be chosen so that $\Pi(\rho_x, P)$ will be arbitrarily small.

G. “If $P = U + S$, where $U$ is countable, and $P$ and $S$ are closed, then $I(P) = I(S)$,” since $I(U) = 0$; then—on the basis of D and G—

H. “The measure of a closed region is equal to the measure of its perfect part.”

We see that Lindelöf’s short communication is extremely rich in content, since it concerns the most important theorems relating to the theory of regions.

43. It will be interesting to note here Borel’s opinion on the significance of Cantor’s works¹:

“When Cantor, twenty years ago, communicated his ideas about counting beyond infinity, they were received not without mistrust. But ana-

lists, astonished by the beauty of his conclusions, did not stop before their somewhat paradoxical form; and then applications of this theory appeared; it also interested those who are interested in the boundaries of mathematics, and Cantor’s ideas became classical for mathematicians and philosophers.”

Somewhat later, in 1903, Borel says¹ that Cantor exerted a considerable influence on the development of mathematics in the last quarter of the nineteenth century, and that this influence will persist, although some of the forms in which Cantor’s thought was cast have acquired only historical interest. By these forms Borel chiefly means the transfinite numbers, whose philosophical significance he does not dispute.

All the theorems that were proved by Cantor and others, including Borel himself, with the aid of transfinite numbers, he considers it desirable to prove by another method, and thinks that this will soon be accomplished.

In this respect the preceding paper of Lindelöf, with its proof of the Cantor–Bendixson theorem, is also of interest.

44. Beginning in 1902, a series of interesting papers by Young began to appear, devoted precisely to the same question as the present investigation, i.e. to the question of the structure and measure of linear domains.²

In the first paper³ the author gives, in connection with Brodén’s investigation, the following construction of a sparsely scattered condensed domain:

The segment $l_1$ is divided by the point $x_1$ into two such parts $l_{01}$ and $l_{11}$ that

\[ \frac{l_{01}}{l_{11}}=\frac{1+j_1}{1-j_1},\qquad j_1=1-\frac{1}{8\cdot 1^2}; \]

then

\[ \tag{1} l_{01}=\left(1-\frac{1}{4^2\cdot 1^2}\right)l_1,\qquad l_{11}=\frac{1}{4^2}l_1. \]

Each of the segments (1) is divided by the points $x_{01}$ and $x_{11}$ into two, $l_{001}, l_{011}$ and $l_{101}, l_{111}$, under the condition that

\[ \frac{l_{001}}{l_{011}}=\frac{l_{101}}{l_{111}}=\frac{1+j_2}{1-j_2},\qquad j_2=-\left(1-\frac{1}{8\cdot 2^2}\right); \]

in such a case

\[ l_{001}=\frac{1}{4^{2}\cdot 2^{2}}\,l_{01}, \qquad l_{011}=\left(1-\frac{1}{4^{2}\cdot 2^{2}}\right)l_{01} =\left(1-\frac{1}{4^{2}\cdot 1^{2}}\right) \left(1-\frac{1}{4^{2}\cdot 2^{2}}\right)l_{1}, \tag{2} \]

\[ l_{101}=\frac{1}{4^{2}\cdot 2^{2}}\,l_{11}, \qquad l_{111}=\left(1-\frac{1}{4^{2}\cdot 2^{2}}\right)l_{11} =\left(1-\frac{1}{4^{2}\cdot 1^{2}}\right)\cdot\frac{1}{4^{2}}\,l_{1}. \]

Each of the four segments (2) is again divided into two parts in the ratio
$\dfrac{1+j_{3}}{1-j_{3}}$, for $j_{3}=+\left(1-\dfrac{1}{8\cdot 3^{2}}\right)$; and so on, at the $n$-th division,

\[ j_{n}=(-1)^{\,n-1}\left(1-\frac{1}{8n^{2}}\right), \tag{3} \]

whereby the largest interval of the $n$-th division will have length

\[ \left(1-\frac{1}{4^{2}\cdot 1^{2}}\right) \left(1-\frac{1}{4^{2}\cdot 2^{2}}\right)\cdots \left(1-\frac{1}{4^{2}\cdot n^{2}}\right)l_{1}. \]

As $n$ increases without bound, this interval has the limit $\dfrac{2\sqrt{2}}{\pi}\,l_{1}$, and its endpoints will be limit points of the sequence $\{x\}$. Similar free intervals will be obtained in each of the intervals of any division, i.e. between any two points of division.

Taking, instead of (3), another law of division,

\[ j_{n}=(-1)^{\,n-1}\left(1-\frac{1}{p^{2}n^{2}}\right), \]

where $p$ is any integer, we would obtain various domains of the type indicated above.

It is obvious that a domain of this kind $\{x\}$, every point of which is a two-sided limit point, will be included among the outer points of the perfect domain determined by the preceding intervals¹.

Young also notes² one oversight by Schoenflies, who says³ that the first example of a nowhere dense perfect domain was given by Smith, whereas only the derived domains of Smith⁴ are perfect; but immediately thereafter Young himself ascribes to Smith the idea and definition of a perfect domain.

45. Young finds¹ that “the study of point regions leads to the investigation of a sequence of intervals, and in certain respects it is more natural to begin with the latter than with the former; this is especially true in the question of the measure of a region”; one cannot but agree with such a statement, and only from this point of view is it possible to establish a comprehensive and rigorously defined notion of measure for a region by means of the construction of the region.

In establishing the fundamental theorems concerning a sequence of intervals, the author has in mind applying them to the investigation of closed regions. The development of his thought, according to his own statement, here proceeds in parallel with Borel, but starts from other considerations. Assigning secondary importance to Borel’s theorem, Young observes that its second “proof is very elegant in idea, but is scarcely capable of explaining to the reader its raison d’être.”

For a sequence of intervals Young distinguishes external and semiexternal (external and semiexternal) points, i.e. two-sided and one-sided limiting points of the region of boundaries, which play a significant role also in our exposition²; the region of semiexternal points must be countable.

Taking a sequence of a finite number $m$ of intervals $\{l_i\}$, Young, like Borel, calls their measure (content) the sum

\[ \tag{4} l=\sum_1^m l_i . \]

If $l$ is less than the length of the fundamental interval $L$, then, besides the intervals $l_i$, there is a sequence of additional intervals $\{\lambda_i\}$, where

\[ l+\lambda=L, \]

where

\[ \tag{5} \lambda=\sum \lambda_i . \]

If $l=L$, then a) there are no additional intervals; b) there are no external points; and c) each boundary, except for the boundaries of the fundamental interval, separates two adjacent intervals from one another.

If the number of non-overlapping intervals $\{l_i\}$ is infinite, then their sequence will, as is known, be countable; in such

the case Young, in agreement with Borel, calls

\[ \sum_{1}^{\infty} l_i = \lim_{n \ldots \infty} \sum_{1}^{n} l_i = l \tag{6} \]

the measure of the series of intervals; it is most convenient to arrange the $\{l_i\}$ in a series according to their magnitude, and we shall always assume this henceforth.

From (6), when the $l_i$ are arranged according to magnitude, it follows that—first—for a given $\sigma$ a number $m_1$ can be found such that

\[ l - \sigma < \sum_{1}^{n} l_i < l \quad \text{for} \quad n > m_1 \tag{7} \]

or

\[ 0 < l - \sum_{1}^{n} l_i < \sigma \quad \text{for} \quad n > m_1, \tag{8} \]

and—second—for a given $\varepsilon < \sigma$

\[ l_n < \varepsilon \quad \text{for} \quad n > m_2. \tag{9} \]

If $m$ is the greatest of the numbers $m_1, m_2$, then simultaneously

\[ 0 < l - \sum_{1}^{n} l_i < \sigma, \qquad l_n < \varepsilon \quad \text{for} \quad n > m. \tag{10} \]

From (10) and (6) we have

\[ 0 < \sum_{n+1}^{\infty} l_i < \sigma, \qquad l_i < \varepsilon, \tag{11} \]

i.e. the sum of those intervals which are smaller than $\varepsilon$ will itself be less than $\sigma$.

If the interval $L$ is divided into a series of other intervals $\{L_j\}$ in such a way that each $l_i$ lies inside one of the $L_j$, then

\[ l = \sum l^{(j)}, \]

where $l^{(j)}$ is the measure belonging to the interval $L_j$.

For a finite number of $L_j$ this is clear in itself; therefore the series $\{L_j\}$ must be assumed infinite. Define $m_1$ according to

condition (8) and, arranging $L_j$ by magnitude, determine $\mu$ so that

\[ L_j < l_{m_1+1}\quad \text{for}\quad j>\mu; \]

then the intervals $l_1, l_2, l_3,\ldots,l_{m_1}, l_{m_1+1}$ can be situated only on the intervals $L_1, L_2,\ldots,L_\mu$; moreover

\[ \sum_1^\mu l^{(j)} \geq \sum_1^{m_1+1} l_i, \]

as a consequence of which—by virtue of (8)—

\[ l-\sum_1^\mu l^{(j)} \leq l-\sum_1^{m_1+1} l_i < \sigma; \]

whence it follows that

\[ l-\lim_{\mu\ldots\infty}\sum_1^\mu l^{(j)}=\sum_1^\infty l^{(j)}. \]

46. If the number of intervals is infinite, then, even when $l<L$, additional intervals do not always exist; likewise the properties indicated above, $b)$ and $c)$, are violated.

Theorem I. If the number of intervals is infinite, there exists at least one limiting point¹.

From this theorem Young derives, as a corollary, Borel’s theorem: having arranged the intervals within which all the points of the given segment $L$ lie in some order—for example, according to magnitude—we shall successively retain only those intervals, or parts of those intervals, which extend beyond the boundaries of the preceding ones; every point of $L$ will then be either an interior point of some interval—possibly a shortened one—or else the boundary of two adjacent intervals; and in such a case, owing to the assumed absence of limiting points, the number of shortened intervals, and consequently also of Borel’s intervals, must be finite.

With regard to this apparently simple proof, it must be observed that it is hardly likely to give greater insight into the meaning of Borel’s theorem than Borel’s second proof.

Theorem II. If, for a sequence of non-overlapping intervals, there are no additional intervals, then every exterior point will be a two-sided limiting point.

This theorem is self-evident.

Among the examples given by the author, let us note one: on the segment $(0,1)$ we take, in the ternary system, the interval $(0.(1),1)$, and then those intervals whose right endpoints are finite fractions expressible only by the digits 0 and 1, while the left endpoints are obtained from the right ones by replacing the last 1 by 0 (1); the isolated intervals thus obtained, with sum equal to 1, determine a perfect domain, whose exterior points are expressed by all the remaining infinite fractions without the digit 2.

This example is a modification of the example B §5 of Smith, whereby each interval of Young is replaced by a countable sequence of adjacent intervals of Smith, and the left endpoints of Young’s intervals turn out to be exterior points of Smith’s intervals.

Modifying another example C §5 of Smith, the author constructs a perfect domain for which $l<1$.

The investigation of the measure of a sequence of Young intervals is reduced to the measure of the isolated intervals determining the perfect domain; in these forms the author constructs a typical ternary sequence of intervals, dividing $(0,1)$ into three parts, taking $\left(\frac{1}{3},\frac{2}{3}\right)$ as the first interval and then applying to the two extreme ones the same process of division.

Then every sequence of isolated intervals may be put into a one-to-one correspondence with the typical sequence, and every property of the exterior points of the former is reduced to the corresponding property of the points of the typical sequence.

Having any sequence of non-overlapping intervals, we can join all adjacent intervals into one; in this case the limiting point of the endpoints of adjacent intervals will become the endpoint of the new interval, or, if this point is the common limiting point of two sequences of adjacent intervals, it will become an interior point of the interval embracing both these sequences.

Continuing such a process, we shall arrive either at a sequence of isolated intervals, or, after a countable sequence of operations, we shall combine all the intervals into one fundamental interval; moreover, at each given moment the sum of the intervals being changed and the sum of the changed intervals remains one and the same; as for the points disappearing during the process of joining intervals, there can be no more than a countable number of them. The final sequence of intervals obtained after

of such a process Young calls the ultimate set. It is evident that this entire process of Young’s is nothing other than a geometric interpretation of the Cantor–Bendixson theorem, applied to the closed domain $P$ of non-inner points, and that the “ultimate set” is the series of free intervals of the perfect domain $P^{(\Omega)}$.

47. To two types of domains, already noted by Lebesgue¹, Young devotes his most interesting investigations. Before passing to them, let us state one lemma²:

A. “Suppose that for every value of $n$ there is a series of intervals
$l_{n1}, l_{n2}, \ldots, l_{n k_n}$, not covering one another and, as $n$ increases, situated on the intervals of the preceding systems, and moreover, for every value of $n$,

\[ s_n=\sum_{i=1}^{k_n} l_{ni}>\lambda; \]

then there exist points lying inside all the intervals $l_{ni}$.”

If the number $k_n$ cannot increase with increasing $n$, the theorem is self-evident, since then at least one of the $l_{ni}$ must be different from zero for every $n$; we therefore suppose that $k_n$ increases together with $n$.

Cutting off from each interval $l_{ni}$, at each of its ends, a part

\[ \frac{\mu}{2^{\,n+1}}\cdot\frac{l_{ni}}{s_n}, \]

where $\mu$ is some small number, we obtain the sum of the cut-off parts no greater than $\dfrac{\mu}{2^n}$, and the sum of the narrowed intervals $l'_{ni}$ no less than $s_n-\dfrac{\mu}{2^n}$.

Take the narrowed intervals $l'_{1i}$; the intervals $l_{2i}$, and those parts of them which fit in $l'_{1i}$, have sum no less than $s_2-\dfrac{\mu}{2}$, so that

\[ \sum l'_i > s_2-\frac{\mu}{2}-\frac{\mu}{2^2}. \]

The intervals $l_{3i}$ and their parts situated on $l'_{2i}$ have a sum not less than $s_3-\dfrac{\mu}{2}-\dfrac{\mu}{2^2}$, and

\[ \sum l'_{3i} > s_3-\frac{\mu}{2}-\frac{\mu}{2^2}-\frac{\mu}{2^3}; \]

and so on; in general

\[ \sum l'_{ni} > s_n-\left(\frac{1}{2}+\frac{1}{2^2}+\cdots+\frac{1}{2^n}\right)\mu > \lambda-\mu . \]

Taking a chain of successive intervals, situated one on another, we determine, as $n$ increases without bound, at least one point $x$ as the limiting boundary of the intervals $l'_{ni}$; this point may be one of the boundaries or an interior point of all the $l'_{ni}$; in either case it is an interior point of the intervals $l_{ni}$; thus the existence of interior points is proved.

The first type is the domain

\[ G=\lim_{n\ldots\infty} G_n, \tag{12} \]

where 1) $G_n$ are closed and rarely scattered domains, and 2) each $G_n$ is contained, as a part, in $G_{n+1}$; thus $G$ is characterized by the circumstance that a) for every value of $n$ all the points of $G_n$ are points of $G$, and b) in $G$ there is not a single point which would not belong to a whole sequence of domains $G_n$, for $n$ greater than some $m$. We suppose, moreover, $G$ to be closed and rarely scattered; the assumption of frequent scattering is of less interest.

Let us construct the free intervals of the domain $G_1$; these intervals, by virtue of the closedness of $G_1$, must be open. Since $G_2$ contains $G_1$, the points of $G_2-G_1$ are situated on the free intervals of $G_1$; likewise, in general, the points of $G_{n+1}-G_n$ lie on the open intervals of the domain $G_n$. Hence it is clear that

B. “Every free interval of the domain $G$ or $G_n$ ascends to some free interval of each of the domains $G_\nu$, for $\nu<n$, either coinciding with it or constituting a part of it; therefore the sum of the intervals of $G$ or $G_n$ cannot be greater than the sum of those intervals to which they ascend.”

C. “For a small positive number $\varepsilon$, an integer $m$ can be determined such that in $G$ and $G_m$ the intervals equal to or greater than $\varepsilon$ will be identical.”

Let $l^{(0)}=(x',x'')$ be a free interval of $G$; then, by virtue of the closedness of $G$, the first such domain $G_m$ can be indicated to which both points $x'$, $x''$ belong; whence it follows that $(x',x'')$ will prove to be a free interval for all $G_n$ with $n>m_1$.

The number of intervals $l_i^{(0)}$ greater than $\varepsilon$ is, of course, finite; for each of them $l_i^{(0)}$, in accordance with the preceding, a number $m_i^{(0)}$ can be determined. If $m'>m_i^{(0)}$, then all intervals of $G$ greater than $\varepsilon$ will enter without change into each of the $G_n$ with $n>m'$.

But into each of these $G_n$ there may also enter other intervals greater than $\varepsilon$, which in the subsequent process have undergone subdivision. The number of such intervals must again be finite; let it be $k_n$. Since in each of them there can be no intervals of $G$ greater than $\varepsilon$, it is possible to indicate for each interval a finite number $\rho_{k_n}$ of points of $G$, standing successively at distances from one another less than $\varepsilon$; since the total number of points $k_n$

\[ \sum_1^{k_n}\rho_{k_n} \]

must be finite, we can find the first domain $G_{m''}$ into which they all enter.

If $m$ is the greater of the numbers $m'$ and $m''$, then $G_n$, for $n>m$, will satisfy the conditions of the theorem. From the preceding it follows that

D. “For $n>m$, all intervals not common to $G_n$ and $G$ will be less than $\varepsilon$.”

The sums of such intervals, less than $\varepsilon$, we shall henceforth denote respectively by $R_n(\varepsilon)$ and $R(\varepsilon)$; it is evident that—in virtue of (6)—always

\[ \lim_{\varepsilon\ldots 0} R(\varepsilon)=0. \tag{13} \]

E. “Conversely: for every given $G_n$ there can be determined such a unique number $\varepsilon_n$ that all intervals of $G_n$ equal to or greater than $\varepsilon_n$ will be contained wholly in $G$.”

For $\varepsilon_n$ we may take the length of the smallest of the common intervals of $G$ and $G_n$, if it is the only one, or, if there are several, but all of them are contained wholly in $G$; if, however, not all such intervals occur in $G$, then for $\varepsilon_n$ we must take the length of the immediately larger interval; in particular cases—for the first domains $G_1, G_2, G_3,\ldots$—the number $\varepsilon_n$ may be set equal to $l$, the length of the whole fundamental

interval, when at first not a single interval $G_1, G_2, G_3,\ldots$ escapes subdivision. But in view of the supposition made at the outset that $G$ is rarely scattered, the appearance of common intervals, in accordance with theorems B and C, becomes necessary.

Hence, in connection with theorem D, for each domain $G_n$ there is a completely definite number $R_n(\varepsilon_n)$, depending on $\varepsilon_n$, which is the sum of the intervals of $G_n$ smaller than $\varepsilon_n$, when all intervals of $G_n$ equal to or greater than $\varepsilon_n$ are adjoined to $G$.

According to the definition of the number $\varepsilon_n$,

\[ \varepsilon_n < \varepsilon_m \quad \text{for} \quad n>m. \tag{14} \]

Since, for every small $\varepsilon$, there may be determined, by virtue of C, a number $m$ and—consequently—a domain $G_m$, for which $\varepsilon_m < \varepsilon$, it follows—in consequence of (14)—that

\[ \lim_{n\ldots\infty} \varepsilon_n = 0. \tag{15} \]

From theorem B it follows that

\[ R(\varepsilon_n) < R_n(\varepsilon_n) < R_m(\varepsilon_m) \quad \text{for} \quad n>m; \tag{16} \]

and in such a case it may be that

\[ \lim_{n\ldots\infty} R_n(\varepsilon_n)=0 \quad \text{or} \quad \ne 0, \tag{17} \]

whereas—by virtue of (13) and (15)—always

\[ \lim_{n\ldots\infty} R(\varepsilon_n)=0. \]

In the first case, for the given small number $\tau$, $m$ can be determined so that

\[ R_n(\varepsilon_n) < \tau \quad \text{for} \quad n>m. \tag{18} \]

F. “If $\lim R_n(\varepsilon_n)\ne 0$, then the domain $G$ cannot be closed.”

Indeed: let

\[ \lim R_n(\varepsilon_n)=L,\quad \text{where always} \quad R_n(\varepsilon_n)>L; \tag{19} \]

by virtue of (19), for an arbitrary $\tau<\dfrac{L}{2}$ there can be found such least numbers
$m_1, m_2, m_3,\ldots,m_n,\ldots$, that

\[ R_{m_1}(\varepsilon_{m_1})-L<\frac{\tau}{2},\quad R_{m_2}(\varepsilon_{m_2})-L<\frac{\tau}{2^2},\ldots,\quad R_{m_n}(\varepsilon_{m_n})-L<\frac{\tau}{2^n},\ldots; \tag{20} \]

exactly likewise, for each $n$ one can find a finite number of intervals $l_{n1}, l_{n2}, \ldots, l_{nk_n}$ such that, for $\sigma < \dfrac{1}{2}$,

\[ \tag{21} \sum_{1}^{\infty} \lambda_{ni} = R_{m_n}(\varepsilon_{m_n}) - \sum_{i=1}^{k_n} l_{ni} < \frac{\sigma}{2^n}, \]

where by $\lambda_{ni}$ are denoted all the remaining, smaller intervals $R_{m_n}$, except for $l_{ni}$.

Hence—in view of (19)—

\[ \tag{22} \sum_{1}^{k_n} l_{ni} > R_{m_n}(\varepsilon_{m_n}) - \frac{\sigma}{2^n} \geq L - \frac{\sigma}{2^n}. \]

If, from the intervals $R_{m_n}$ that are smaller than $\varepsilon_{m_n}$, we shall pass to the corresponding intervals $G_{m_s}$, with $m_s < m_n$ and—consequently—with $\varepsilon_{m_s} \geq \varepsilon_{m_n}$, then we can encounter them only among the intervals $R_{m_s}$, since the remaining intervals $G_{m_s}$, equal to or greater than $\varepsilon_{m_s}$, inevitably enter into $G$.

Having prefaced these remarks, let us call the intervals $R_{m_1}$ to which the intervals $l_{2i}$ ascend, $l_{2i}$; then—in view of B and (22)—

\[ \sum l_{2i} > \sum l_{2i} \geq L - \frac{\sigma}{2^2} > \frac{\sigma}{2}. \]

Among the intervals $l_{2i}$, coinciding with various intervals $R_{m_1}$, some $l'_{2i}$ enter into the composition of $l_{1i}$, while the others $l''_{2i}$ belong to the number of intervals $\lambda_{1i}$; the intervals $R_{m_1}$ ascending to them we shall denote respectively by $l'_{2i}$ and $l''_{2i}$. Since the entire sum of the intervals $\lambda_{1i}$—in view of (21)—is less than $\dfrac{\sigma}{2}$, it must all the more be that

\[ \tag{23} \sum l''_{2i} < \sum \lambda''_{2i} < \frac{\sigma}{2}. \]

But—in view of (22)—

\[ \sum l'_{2i} + \sum l''_{2i} = \sum_{1}^{k_2} l_{2i} > L - \frac{\sigma}{2^2}; \]

therefore—in connection with (23)—

\[ \tag{24} \sum l'_{2i} > L - \left(\frac{1}{2} + \frac{1}{2^2}\right)\sigma, \]

and all the more

\[ \sum l'_{2i} > L-\left(\frac{1}{2}+\frac{1}{2^2}\right)\sigma, \tag{25} \]

i.e. the sum of those intervals $l_{2i}$ which ascend to the intervals $l_{1i}$, and the sum of these latter intervals, exceed $L-\left(\frac{1}{2}+\frac{1}{2^2}\right)\sigma$.

Since

\[ R_{m_2}(\varepsilon_{m_2})=\sum l'_{2i}+\sum l''_{2i}+\sum \lambda_{2i}, \]

then—in view of (24) and (20)—

\[ \begin{aligned} \sum l''_{2i}+\sum \lambda_{2i} &=R_{m_2}(\varepsilon_{m_2})-\sum l'_{2i} < R_{m_2}(\varepsilon_{m_2})-L+\left(\frac{1}{2}+\frac{1}{2^2}\right)\sigma \\ &< \frac{\sigma}{2^2}+\left(\frac{1}{2}+\frac{1}{2^2}\right)\sigma . \end{aligned} \tag{26} \]

Next: the intervals $l_{3i}$ ascend to the intervals $G_{m_2}$ and $G_{m_1}$. All intervals $R_{m_3}$ split into two parts $l_{3i}$ and $\lambda_{3i}$, the former again into $l'_{3i}$ and $l''_{3i}$; the $l'_{3i}$, through the intervals $l'_{2i}$, ascend to the intervals $\bar l'_{2i}$ included in

\[ \sum_{1}^{k_1} l_{1i}, \]

whereas the $l''_{3i}$ ascend to the intervals $R_{m_1}$ through the intervals $l''_{2i}$ and $\lambda_{2i}$.

In this case—in view of (26)—

\[ \sum l''_{3i} \leq \sum l''_{2i}+\sum \lambda_{2i} < \frac{\sigma}{2^2}+\left(\frac{1}{2}+\frac{1}{2^2}\right)\sigma; \]

but—on the basis of (22)—

\[ \sum l'_{3i}+\sum l''_{3i} =\sum_{1}^{k_3} l_{3i} > L-\frac{\sigma}{2^3}, \]

therefore

\[ \sum l'_{3i} > L-\frac{\sigma}{2^3}-\sum l''_{3i} > L-\frac{\sigma}{2^2} -\left(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}\right)\sigma, \tag{27} \]

and all the more

\[ \sum l'_{3i} > L-\frac{\sigma}{2^2} -\left(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}\right)\sigma; \]

whence—in virtue of (27) and (20)—

\[ \begin{aligned} \sum l_{3i}^{\prime\prime}+\sum l_{3i} &=R_{m_3}-\sum l_{3i}^{\prime} < R_{m_3}-L+\frac{\tau}{2^2} +\left(\frac12+\frac{1}{2^2}+\frac{1}{2^3}\right)\sigma \\ &< \left(\frac12+\frac{1}{2^2}+\frac{1}{2^3}\right)\sigma +\left(\frac{1}{2^2}+\frac{1}{2^3}\right)\tau ; \end{aligned} \tag{28} \]

and so on; in general, the sum of those intervals $l'_{ni}$ which, through all the intermediate intervals $l'_{n-1,i},\, l'_{n-2,i},\ldots, l'_{2i}$, ascend to $l_{1i}$, will be

\[ \sum l'_{ni}>L-\left(\frac12+\frac{1}{2^2}+\cdots+\frac{1}{2^n}\right)\sigma -\left(\frac{1}{2^2}+\frac{1}{2^3}+\cdots+\frac{1}{2^{n-1}}\right)\tau ; \]

all the more, for every $n$,

\[ \sum \bar l'_{ni}\geq \sum l'_{ni}>L-\sigma-\tau ; \tag{29} \]

the sums of the intermediate intervals through which the ascent is made satisfy the same inequality—in virtue of B; and in such a case—in virtue of Lemma A—there exist points which lie inside all the $l'_{ni}$ and serve as limit points of their endpoints; these points do not belong to the domain $G$, which thus turns out to be non-closed.

From Theorem F it follows that

G. “If the domain $G$ is closed, then

\[ \lim R_n(\varepsilon_n)=0”; \]

whence—in virtue of (18)—we have

H. “If $G$ is closed, then, for small $\sigma$, a number $m$ can be determined such that

\[ R_n(\varepsilon_n)<\sigma \quad \text{for } n\geq m, \]

i.e. for $G$ and $G_n$ all intervals equal to or greater than $\varepsilon_n$ will be the same, and then the sum of the intervals smaller than $\varepsilon_n$ will be less than $\sigma$.”

Young adds two further remarks: if $G$ is not closed, and $\overline{G}$ is the domain obtained from it by closure, then

\[ \overline{G}=M\{G,\,G'\}. \]

In the case where $\overline{G}$ is perfect, $G$ must be dense, and $G'=\overline{G}$; but if $\overline{G}$ is not perfect, $G'$ cannot be identical with $G$; indeed—from $G'=\overline{G}$ it would follow that $G$ is dense and $G'=\overline{G}$ perfect, which cannot be.

If $\bar G$ is perfect, and—therefore—$G$ is dense but not closed, then from the definition $G=\lim G_n$ one still cannot infer that

\[ \lim G'_n = G' = \bar G; \]

indeed: since—by hypothesis—$G_n$ is closed, $G'_n = D(G_n)$; consequently $\lim G'_n = D(G)$, whereas $G'$ contains points not belonging to $G$.

From Theorem H there follows directly the theorem of Osgood¹:

J. “If $G$ is closed, then

\[ \mathrm J(G)=\lim_{n\ldots\infty}\mathrm J(G_n). \tag{30} \]

Indeed: defining the measure of $G_n$ and $G$ as

\[ \mathrm J(G_n)=l-\sum_1^\infty l_{ni};\qquad \mathrm J(G)=l-\sum_1^\infty l_{0i}, \]

where $l_{ni}, l_{0i}$ are the free intervals of these domains, we have—according to H and (16)—

\[ \left|\mathrm J(G_n)-\mathrm J(G)\right| = \left|\sum_1^\infty l_{0i}-\sum_1^\infty l_{ni}\right| = \left|\mathrm R(\varepsilon_n)-\mathrm R_n(\varepsilon_n)\right|<2\sigma, \]

whence (30) follows.

If $G$ is not closed, let us determine² the most general conditions under which

\[ \lim \mathrm J(G_n)=\mathrm J(G) \tag{31} \]

In this case

M. “For arbitrarily given $\varepsilon$ and $\sigma$ there can be determined a number $m$ such that, for $G_n$ and $G$, when $n>m$, the sums of free intervals equal to or greater than $\varepsilon$ differ from one another by less than $\sigma$.”

This theorem is a generalization of Theorem H, where $G'=G$, and where the intervals here and there coincided completely.

Let $(x', x'')$ be one of the intervals of $G$ which is $> \varepsilon$, and let the number of such intervals be $k$; lengthening it at each end by $\frac{\sigma}{2k}$, we obtain the points $\bar{x}'_0,\bar{x}''_0$; on these small segments there lie arbitrarily many points of $G$; indeed: if the points $\bar{x}',\bar{x}''$ are isolated points of $\bar G$, then they must belong to $G$, and then the process of lengthening is rendered useless; if, however, one or both of the points $\bar{x}',\bar{x}''$ are limiting points, then in their neighborhoods there are arbitrarily many points of $G$. Thus it is always possible to choose on $(\bar{x}'_0,\bar{x}')$ and on $(\bar{x}'',\bar{x}''_0)$ points $x',x''$ which will be points of $G$ and—consequently—points of some $G_{m_1}$. Then the free interval $(x',x'')$, or that part of it on which $(\bar{x}',\bar{x}'')$ is situated, will certainly belong to $G_n$ for $n>m_1$. Defining such numbers $m^{(i)}_1$ for each of the $k$ intervals and taking $m_2$ to be the greatest of them, we obtain that, for $n>m_2$, the sums of the free intervals of $G_n$ and $\bar G$ differ from one another by a quantity less than $\sigma$.

But in $G_n$ there may be intervals equal to or greater than $\varepsilon$ which are not included in $G$, and whose number is finite; with them we can proceed in the same way as in the proof of theorem C, and we then obtain a number $m>m_2$ satisfying the conditions of the theorem.

We have, therefore, for $n>m$,

\[ \left| \left\{\sum_{i=1}^{\infty} l_{n_i}-R_n(\varepsilon)\right\} - \left\{\sum_{i=1}^{\infty} l_{n_i}-R(\varepsilon)\right\} \right|<\sigma \]

or

\[ \left|\{\,l-J(G_n)-R_n(\varepsilon)\,\}-\{\,l-J(G)-R(\varepsilon)\,\}\right| = |\,J(G)-J(G_n)-R_n(\varepsilon)+R(\varepsilon)\,|<\sigma; \]

since—by virtue of D—$\lim_{\varepsilon\ldots 0} R(\varepsilon)=0$, then, in order that (31) be satisfied, it is necessary and sufficient that $R_n(\varepsilon)$ be arbitrarily small for sufficiently small $\varepsilon$ and large $n$.

To explain the whole preceding theory Young gives the following example: on the segment $(0,1)$ construct 1) in the ternary system the closed first Smith series, calling it $G_1$, and 2) $G$—the closed second Smith series¹; they will have the common free interval $(0.2,1)$. Since the series in the intervals $(0,0.1)$ and $(0.1,0.2)$ are identical, we shall consider only the first of them. On the three largest

in the free intervals of the set $G_1$ on $(0,0.1)$ we again place the first Smith series $a$; then on $(0,1)$ there is obtained a domain $G_2$, in which there will be in common with $\Gamma$ three intervals equal to or greater than $\dfrac{1}{3^{1+2}}$. The series on the sixteen intervals $(0,0.1)$ and $(0.1,0.2)$ are identical, and we shall consider only $(0,0.001)$; in the three largest intervals of the domain $G_2$ on $(0,0.001)$ we again place the first Smith series $a$, obtaining thereby on $(0,1)$ the domain $G_3$, and then with the $3^2$ largest intervals of $G_3$ on $(0,0.001)$ we do the same, obtaining on $(0,1)$ the domain $G_4$; this domain has in common with $\Gamma$ intervals equal to or greater than $\dfrac{1}{3^{1+2+3}}$, and so on. We shall obtain in this way a domain $G$ as the limit of the $G_n$, and for a given $\varepsilon$ it may be possible to choose $m$ so that the free intervals of $G_n$ and $\Gamma$, equal to or greater than $\varepsilon$, will be one and the same; closing $G$, we obtain a set $G$ which will coincide with $\Gamma$.

The sum of the free intervals of $G_n$ is equal to 1, and $\mathrm J(G_n)=0$; consequently also $\lim \mathrm J(G_n)=0$; whereas $\mathrm J(\Gamma)=\mathrm J(G)$ lies between $\dfrac{1}{2}$ and $\dfrac{2}{3}$. Thus, despite the fact that all intervals equal to or greater than $\varepsilon$ are common to $\Gamma$ and $G_n$, the difference of the sums of the remaining smaller intervals $R_n(\varepsilon_n)-R(\varepsilon_n)$ lies between $\dfrac{1}{2}$ and $\dfrac{1}{3}$.

Much space is devoted in Chapter II of the present exposition to domains of this type.

48. Young further notes¹ that the notion of dispersion of one domain over another has not been sufficiently established.

If $P$ is part of a closed, rarely dispersed domain $Q$, then Schoenflies² calls $P$ partly dispersed over $Q$, if

\[ P' = Q. \tag{32} \]

Young considers this definition unsatisfactory, since it may happen that no part of the domain $Q$ has $Q$ as its derivative; this will always occur for a closed but not perfect $Q$. In view of this, Young proposes to take as the definition of partial dispersion the condition

\[ Q = M(P,P'). \tag{33} \]

To clarify by an example the difference between that definition and the other, let us take the series

\[ \tag{34} Q=\left\{0,\ldots,\frac{1}{2^n},\ldots,\frac{1}{2^3},\frac{1}{2^2},\frac{1}{2},1\right\}; \]

according to Schoenflies’ definition, there is no nowhere-dense series over such a domain; according to Young, the series

\[ P=\left\{\ldots,\frac{1}{2^n},\ldots,\frac{1}{2^3},\frac{1}{2^2},1\right\}, \]

which contains all isolated points of $Q$, will be the only one nowhere dense with respect to the series $Q$.

It is evident that, according to Young, the presence in $P$ of all isolated points of $Q$ is obligatory; indeed: since $Q$ is, by hypothesis, closed,

\[ Q=Q_i+Q', \]

where $Q_i$ is the domain of isolated points of $Q$; since $P$ is a part of $Q$, $P'$ consists of limiting points of $Q$, i.e. $P'=D(Q)$; consequently $M(P,P')$ contains only those isolated points of $Q$ which are in $P$. Therefore condition (33) can be fulfilled only in the case when all points $Q_i$ are included in $P$.

In view of this, it might perhaps be desirable to define a nowhere-dense domain so that this definition, being a development of the corresponding definition for the continuum, would give greater freedom in the construction of nowhere-dense domains.

Such a definition could be the following: a part $P$ of a closed domain $Q$ is called nowhere dense with respect to $Q$ if

\[ \tag{35} P'=Q'; \]

in this case there is no obligation to include in $P$ all isolated points of $Q$.

According to this definition, for the example (34), nowhere-dense domains with respect to $Q$ will be, for instance, the domains

\[ \left\{\ldots,\frac{1}{2^{kn}},\ldots,\frac{1}{2^{3k}},\frac{1}{2^{2k}},\frac{1}{2^k},\frac{1}{2^0}\right\}, \]

and so on, for every integer value of $k$.

It is evident that, under definition (35), among the isolated points only such domains of them must enter into $P$ that not a single limiting point be lost. For perfect domains $Q$, all three definitions coincide.

A third definition of a part of a scattered domain is the finding of such domains as a particular case of a more general problem, namely, the reconstruction, from the derivative $P' = Q'$, of the original domain $P$, a problem first posed by Bendixson alone¹.

This problem, like indefinite integration in general, has infinitely many solutions; among these solutions, the parts of the scattering of the domain $P$ that are components of $Q$ answer to the definition; this imposes an additional condition on the choice of $P$.

49. Studying further² domains made up of intervals that overlap one another, Young admits, in contrast to the conditions of Borel’s theorem, the possibility of the existence of points of a segment that are not internal for all intervals of the domain. These non-internal points may belong to three categories: 1) points serving as boundaries of two or several adjacent or mutually superposed intervals, 2) external points serving as two-sided limiting points of rows of boundaries, and 3) one-sided limiting points. A row of intervals determined by these non-internal points Young calls an equivalent row of intervals not overlapping one another. From this definition it follows that

N. “Every internal point of the intervals of an equivalent row will necessarily be internal for at least one interval of the given row, and conversely.”

Using this concept, Young establishes a theorem representing a generalization of Borel’s theorem:

O. “If every point of the closed domain $P = \{x\}$ lies inside at least one of the intervals $\{l_i\}$, then it is possible to determine a finite number of intervals with this property.”

Let $\{L_i\}$ be an equivalent row of intervals; since every point $x$ lies inside $\{l_i\}$, then, by virtue of N, it will also be contained inside at least one of the $L_i$. The number of intervals $L_i$ must be finite, since otherwise there would exist for the boundaries of the intervals $L_i$, and consequently also for the points $x$ lying on them, a limiting point which 1) would not lie inside $\{L_i\}$ and 2) would be a point $x$, since $\{x\}$ is a closed domain; this contradicts the definition of $\{l_i\}$ and theorem N. Let

\[ L_1,\ L_2,\ \ldots,\ L_k \tag{36} \]

be the intervals of the equivalent row inside which the points $\{x\}$ lie.

In that case, on each of the intervals (36) we can construct an interval $L'_j=(\xi_j,\eta_j)$ so that the $\{x\}$ will lie inside $L'_j$; since, by construction, $\xi_j,\eta_j$ will not be points $\{x\}$, the part $P'_j$ of the set $P$ situated on $L'_j$ will be closed. In view of the fact that every point of $L'_j$, including all the points of $P'_j$, will be an interior point of the interval $L_j$, it—by virtue of $N$—will be interior also for $\{l_i\}$. Therefore one can indicate a finite number $n_j$ of intervals $\{l_i\}$ possessing this same property. Taking such intervals $l_i$ for all $L_j$, we obtain a finite number

\[ \sum_{j=1}^{k} n_j=n \]

of intervals $\{l_i\}$, which will possess the required property with respect to the points of the given set.

At the same time it is obvious$^{1}$ that it is impossible to define an infinite sequence of non-overlapping intervals so that: 1) every point of a closed set should lie inside one of them, and 2) there should not be a single solid interval free of points of the set.

50. Young$^{2}$ defines the measure $J(P)$ of a closed point set as $l-J_1$, where $J_1$ is the sum of its free intervals. From this definition the old definition follows immediately:

P. “If the points of a closed set are enclosed inside a finite number of non-overlapping intervals, then the sum of these intervals is greater than $J(P)$, but can be made arbitrarily close to $J(P)$.”

Indeed: let us take an as yet undetermined $\varepsilon$ and determine a finite number of free intervals $l_i$ which are greater than $\varepsilon$; then there also remains a finite number $k$ of intervals $\lambda_j$, on which, as boundary points and interior points, all the points of the set $P$ lie, and besides on $\lambda_j$ lie all the free intervals $l_i<\varepsilon$, whose sum let be $R(\varepsilon)$. If each of the $\lambda_j$ is lengthened at each of its ends by a quantity less than $\dfrac{\sigma}{2k}$, then all the points of $P$ will already lie inside the intervals $\lambda'_j$ thus obtained, which will partly cover the intervals $l_i$; their common part will be less than $\sigma$. Hence the measure of those parts of the free intervals which—

$^{1}$ ib. (2) 1, p. 233.
$^{2}$ ib. p. 232.

which lie outside $\lambda'_j$, will be contained between $J_l - R(\varepsilon)-\varepsilon$ and $J_l - R(\varepsilon)$, and consequently

\[ J(P)+R(\varepsilon)=l-\lvert J_l-R(\varepsilon)\rvert < \sum_{1}^{k}\lambda'_j < l-\lvert J_l-R(\varepsilon)-\varepsilon\rvert = J(P)+R(\varepsilon)+\varepsilon, \]

i.e.

\[ R(\varepsilon)<\sum \lambda'_j-J(P)<R(\varepsilon)+\varepsilon; \]

but the sum $R(\varepsilon)$, for sufficiently small $\varepsilon$, can be made arbitrarily small, whence the assertion of the theorem follows.

By this theorem one of the categories of definitions of measure is reduced to Young’s definition; in exactly the same way, for the other category, the author gives the following theorem:

Q. “If every point of a closed domain is made the midpoint of a small interval, then the sum of a finite number of non-overlapping intervals, which are filled by the preceding intervals, as their lengths decrease indefinitely, will have $J(P)$ as its limit.”

Indeed: observing that—on the basis of theorem O—a sequence of small overlapping intervals may be replaced by a finite number of them, we can reduce theorem Q to theorem P.

Young’s definition of measure and these two theorems finally establish that order which had long been insistent; now it remains only to ascertain to what extent Young’s definition can be extended to non-closed domains.

It is clear that a closed sequence with measure $l$, equal to the length of the fundamental interval, must necessarily coincide with the continuum, since for it one must have $J_l=0$.

Starting from those intervals by which the point domain $P$ is defined, Young clarifies the geometric origin of the derived domain $P'$: passing from the free intervals of $P$ to the corresponding intervals of $P'$, we join together all adjacent intervals up to their limiting points on both sides; if two countable sequences of intervals have a common limiting point which is external with respect to both the one and the other intervals, then this point will now be the boundary of the free adjacent intervals of the domain $P'$.

The finding of further derived domains $P''$, $P'''$, … will have as its consequence a new joining of intervals into one; in this process

successively the one-sided and two-sided limiting points disappear; the number of points that have disappeared can only be countable.

If the domain $P$ is countable, then after a certain series of operations the free intervals will merge into one continuous fundamental interval; if, however, it is not countable, then we obtain the ultimate set of the isolated intervals: the perfect domain $P^{(2)}$ of the non-internal points of these intervals. Young calls the nucleus or kernel (nucleus) of the domain $P$.

In this form the fundamental theorem of Cantor-Bendixson is presented by the author; it is evident $^{1}$ that

The kernel $P^{(2)}$ will have the same measure as $P$, and also the same external points, except—possibly—for a countable set.

Young further observes that one often has, given a countable series of domains

\[ P_1,\ P_2,\ P_3,\ldots,\ P_n,\ldots,\quad \text{with}\quad P_{n+1}=D(P_n), \]

to speak of all points common to all the domains $P_i$; the domain of these points, $P_\omega$, he calls the derived (deduced) domain, and the process itself—deduction.

“If a finite or countable series of operations for finding derivatives and their derivatives is given, one can construct a countable domain of points which disappears if these operations are performed on it.”

This assertion has a certain analogy with the theory of distribution developed by us in the second chapter.

51. Young sets himself the aim $^{2}$ of investigating an even more generalized domain of Borel $^{3}$: we make each of the points of the domain $P$ the midpoint of some interval; causing all the intervals $l_i^{(n)}$ to tend to zero with increasing $n$, we investigate the domain of internal points of these intervals. With respect to this domain, which Young calls the inner limiting set, he gives the following theorems.

R. “Every point $x^{(0)}$ of the inner limiting set $P^{(0)}$ will be a point of $P$.”

Indeed: $x^{(0)}$ must be an internal point for each set of intervals $\{l_i^{(n)}\}$ for every $n$; the intervals containing inter—

$^{1}$ Cf. 17°.

$^{2}$ Proceedings, (2) 1, p. 262.

$^{3}$ Cf. 30°.

together with themselves, besides $x^{(0)}$, still infinitely many points $P$ vanish. Consequently, $x^{(0)}$ must be a point $P'$.

From this it follows directly that

D. “For a closed domain $P$, always $P = P^{(0)}$.”

S. “$P^{(0)}$ may contain every point $P'$.”

Let us take all the intervals $i_i^{(n)}$, for a given $n$, equal; then every point $P'$ will be interior to

\[ \sum_1^\infty i_i^{(n)} \]

for every $n$, i.e. it will enter into $P^{(0)}$; but, generally speaking, $P'$ need not be $D \mid P^{(0)}$.

T. “If $P$ is a separable domain, and $P'$ is countable$^*$, then the intervals can be placed so that $P^{(0)} = P'$.”

On the basis of R—from the points not entering into $P$, only points $P'$ can conclude in $P^{(0)}$. The points $P'$ which belong to $P$ raise no doubt here; it remains only to prove that the intervals can be arranged so that none of them will contain any point $P'$ not included in $P$.

Let us take the closed domain

\[ P = M(P, P'); \]

the isolated points $\{x_{0i}\} = P_a$ will be identical with the isolated points $P$; let $d_{0i}$ be the distance of the point $x_{0i}$ to the nearest point $P_a$. Then all limiting points of $P$ will not lie inside the intervals

\[ i_{0i}^{(n)}=\left(x_{0i}-\frac{1}{2^n}d_{0i},\,x_{0i}+\frac{1}{2^n}d_{0i}\right). \]

After removing from $P$ the points $P_a$, we obtain $P_a$—a domain of isolated points of the domain $P-P_a$; some of these points $\{x_{1i}\}$ enter into $P$, while others $\{\xi_{1j}\}$ do not enter there. The domain $\{x_{1i},\xi_{1j}\}$ will again be isolated, and about its points one may construct intervals

\[ i_{1i}^{(n)}=\left(x_{1i}-\frac{1}{2^n}d_{1i},\,x_{1i}+\frac{1}{2^n}d_{1i}\right);\quad \lambda_{1j}^{(n)}=\left(\xi_{1j}-\frac{1}{2^n}\delta_{1j},\,\xi_{1j}+\frac{1}{2^n}\delta_{1j}\right), \]

inside each of which 1) there will be no other points $P_a$, except $x_{1i}$ or $\xi_{1j}$, and—consequently—the more so no other points $P$, which serve as limiting points for $x_{1i}, \xi_{1j}$, and 2) all limiting points $P_a$ do not lie inside $i_{1i}^{(n)}$ and $\lambda_{1j}^{(n)}$; $d_{1i}$ and $\delta_{1j}$ are here again the distances of $x_{1i}$ or $\xi_{1j}$ to the nearest point $P_a$.

$^*$ i.e. $P$ is reducible.

The isolated points of the domain $P-P_a-P_{c a}$ give the domain $P_{c a}$, and only the points $\{x_{2j}\}$ entering into it may belong to $P$, whereas the others $\{\xi_{2j}\}$ may not.

Constructing around them intervals $l_{2i}$ and $k_{2i}$, and continuing this process further, we shall include the points of $P$ in intervals which, generally speaking, contain one another,

\[ \{l_{0i}^{(n)}\},\quad \{l_{1i}^{(n)}\},\quad \{l_{2i}^{(n)}\},\ldots,\{l_{\chi i}^{(n)}\}, \]

where $\chi$ is a number of the second class at which the preceding process must break off, since the domain $P$ is reducible; for sufficiently large values of $n$, not a single point $\{\xi_{kj}\}$ lies inside these intervals; the point $\xi_{kj}$ may lie inside the intervals $l_{k+1,i}^{(n)}, l_{k+2,i}^{(n)},\ldots$, but, as $n$ increases, it must find itself outside these intervals, since $\xi_{kj}$ lies at a finite distance from each of the points $x_{k+1,j}, x_{k+2,j},\ldots$.

Giving $n$ all values of the natural series $1,2,3,\ldots$, we shall obtain a domain $P^{(0)}$, into whose composition there will not enter a single point of $P'$ not belonging to $P$; and in that case—according to K—it must be that $P^{(0)}=P$.

U. “If $P$ is everywhere dense and is not separable, $P^{(0)}$ has the power of the continuum.”

Indeed: since $P$ is not separable, $P^{(0)}=M(P)$ cannot be separable either; and in such a case $P^{(0)}$ is not countable².

V. “If $P$ is a separable domain, but $P'$ is not countable, $P^{(0)}$ may either be countable or have the power of the continuum; moreover it is possible to place the intervals so that $P^{(0)}=P$.”

If, for an uncountable $P'$, the domain $P$ is separable, then the points of $P$ will be distributed over the free intervals of some perfect domain $Q$, and in each of the intervals a domain of the first kind will be placed.

Hence there is the possibility of constructing, in each free interval of $Q$, intervals of type $T$ in such a way that it turns out that $P^{(0)}=P$, i.e. $P^{(0)}$ will be countable, as is $P$.

If, on the contrary, intervals of type $S$ are constructed around the points $P$, then $P^{(0)}$ will contain in itself the perfect region $Q$, and will therefore have a discontinuity gap.

W. “In general it is possible to place the intervals in such a way that the points $P^{(0)}$ not belonging to $P$ will be limit points of the condensed part $P$.”

Indeed: constructing, for the points of the condensed part $P$ often scattered over a certain perfect region $Q$, intervals of type $S$, and for the points lying inside the free intervals of $Q$, intervals of type $T$, we obtain

\[ P^{(0)} = M\{P,Q\}. \]

Besides those indicated in the text, Young gives one more theorem¹:

X. “If the measure of the intervals $\lambda^{(u)}$ containing $P^{(0)}$ can be made smaller than the measure of $P'$, then the countable part $D(P')$ is not contained in $P^{(0)}$.”

Let $D(P')=\{x_i\}$ be countable, and let $J(P')-u\mathrm{Gr}.\,\lambda^{(u)}=2\varepsilon$; construct around each point $x_i$ an interval

\[ \left(x_i-\frac{\varepsilon}{2^{i+1}},\; x_i+\frac{\varepsilon}{2^{i+1}}\right); \]

then the measure of the region $\{P^{(0)},D(P')\}$, containing all the points $P'$, will be no greater than $u\mathrm{Gr}.\,\lambda^{(u)}+\varepsilon$, i.e. smaller than $J(P')$, which is impossible; consequently, one cannot suppose the region $D(P')$ to be countable.

Analogously one can prove that

Y. “If the region $P_1$ constitutes a part of $P_2$, and the measure of $P_1$ is smaller than the measure of $P_2$, then the region $P_2-P_1$ has a discontinuity gap.”

52. The proof² of the Cantor–Bendixson theorem is based on the notion of $\Omega$, the first number of the third class; moreover, this number, as Cantor himself notes, does not enter at all into the final result. Therefore it was natural that Schoenflies should wish to give³ such a proof of this theorem as would be entirely independent of $\Omega$.

Schoenflies’s proof is presented in the spirit of Young, who himself made⁴ an indication of this. It is based on the fact that

a) to every number of the second class there corresponds the type of arrangement of a certain precisely arranged¹ countable domain, and conversely; and b) every precisely arranged domain of positive decreasing numbers is countable, and its type of arrangement is a certain number of the second class.

By virtue of the definition, the numbers of a domain may be given in the form

\[ a_1 > a_2 > a_3 > \ldots > a_\omega > a_{\omega+1} > \ldots > a_\alpha > a_{\alpha+1} > \ldots ; \tag{1} \]

if we denote $a_\nu - a_{\nu+1} = b_\nu$, then

\[ b_1,\ b_2,\ b_3,\ \ldots,\ b_\omega,\ b_{\omega+1},\ \ldots,\ b_\alpha,\ b_{\alpha+1},\ \ldots \tag{2} \]

will be another sequence in which all the numbers $b_\nu$ are finite and positive, and the sum of an arbitrary number of numbers (2) does not exceed $a_1$. Therefore, if one takes any sequence of infinitely decreasing numbers $\{\varepsilon_i\}$, then between any two numbers $\varepsilon_i$ there can lie only a finite number of the numbers $b_\nu$; whence follows the countability of the sequences (2) and (1).

In the proof of the Cantor–Bendixson–Schoenflies theorem one starts from the sequence $\{l_i\}$ of intervals determining a closed domain, which, after a countable number of operations, must yield either a continuous fundamental interval or a sequence of isolated intervals; the proof remains valid also in the case when the given domain consists of a series of sparsely and frequently scattered parts.

Then, by the same method, the author proves also Baire’s theorem²:

“If $Q$ is a closed domain, and

\[ Q_1,\ Q_2,\ Q_3,\ \ldots,\ Q_\omega,\ Q_{\omega+1},\ \ldots,\ Q_\alpha,\ Q_{\alpha+1},\ \ldots \]

is a sequence of domains, of which $Q_1$ is a closed part of $Q$, and in general each $Q_{\rho+1}$ is a closed part of $Q_\rho$, then there exists a least number $\alpha$ of the first or of the second class for which $Q_\alpha$ is empty or perfect.”

Both proofs are distinguished by extreme simplicity and persuasiveness.

The author's later works¹ on the theory of point domains, as well as the article by Zoretti², concern two-dimensional domains and, relying on new methods and new concepts, lie outside the circle of ideas of the present investigation.

53. Having followed step by step how the new branch of pure mathematics developed, how it worked out its methods and propositions, we shall now try to present the general picture of this development and to indicate the directions in which its further elaboration should proceed.

In the theory of domains, which arose and developed almost before our eyes, it is interesting to trace the history of a new discipline, a history which, in miniature, repeats the history of the development of science in general. The beginnings of every science are lost in the darkness of time, and the separate moments of its development are marked by centuries, whereas in the theory of domains the whole cycle passes within an extraordinarily short period of time.

From isolated facts in Bolzano, curious as “paradoxes,” the doctrine of domains acquires in Cantor already the form of a theory; but the extent to which this theory was distinctive is indicated already by the circumstance that Cantor for a long time did not decide to publish his investigations. In the first period after their publication there were very few persons who occupied themselves with this new branch of mathematics, but as time went on the number of workers in the field of Mengenlehre became ever larger, its influence penetrated into various divisions of science, and it began already to enter elementary courses³.

Thus the right to existence and the role of the doctrine of domains in the general system of science is consolidated; this doctrine is reckoned with, and at the present time one can no longer avoid its influence in a whole series of separate divisions of analysis. And this entire evolution took place in the course of some 30 years, not counting its so-called prehistorical period.

It is then interesting to trace how various ideas of the doctrine of domains constantly arise, independently of one another, among different investigators; this phenomenon is so frequent that comparatively little takes its origin from any one person, and for the greater part the concepts have to be traced back to two, or even more, authors.

Further, the part played by each of the individual investigators in the development of the theory usually proves to be very small: often some single example, one theorem, or one concept exerts an essential influence on the whole subsequent course of development. Apart from Cantor himself, only Borel and Young have more extensive investigations in the theory of linear sets, and Jordan and Schoenflies in the theory of plane domains.

54. The doctrine of domains (Mengenlehre) split first of all into two currents: one occupied itself with transfinite arithmetic and the theory of transfinite numbers, while the other turned its attention chiefly to point domains; this latter is, to a certain extent, unfriendly toward transfinite numbers, and there is an ever stronger tendency to found the theory of point domains independently of them; the representative of this current is Borel.

It is hardly possible to expect complete success in this direction: if it proves possible to found the theorems concerning point domains independently of $\Omega$—the first number of the 3rd class—then, one must think, numbers of the 2nd class will always retain their proper significance.

Despite their close connection, the theory of transfinite numbers and the theory of point domains each have a special circle of workers, who only rarely and reluctantly go beyond the limits of their own theory; and mathematicians such as Borel and Schoenflies, who have devoted attention to both branches of the doctrine of domains, direct their efforts toward separating them from one another and welcome the successes of others in this field.

The theory of point domains, to which the present investigation is devoted, has as its task the study of a) the size of a domain, b) its structure, and c) its measure.

As regards size, Cantor had already established that point domains may be finite, countable, or have the power of the continuum. But the clarification of the question of how this last power is related to the sequence of alephs belongs to the tasks of the theory of transfinite numbers and does not concern us here.^1)

The study of the structure and measure of a domain proceeded, for the most part, likewise to a certain degree independently of one another, and mathematicians who occupied themselves with one paid little attention to the other; in this respect an especially characteristic example is G. Cantor—the creator of the doctrine of domains, who was comparatively little interested in the question of measure.

^1) In the recently published booklet Jahresber. d. Deut. Math.-Ver. (vol. XIV, p. 447), Bernstein reports that he has succeeded in establishing that this power is $\aleph_1$.

55. Bolzano, to whom the entire doctrine goes back, touched upon all three questions that made up the content of the theory of point domains.

Then, before Cantor, the attention of analysts was drawn to establishing the condition of integrability, and in connection with this the question of measure began to develop.

Dividing the fundamental interval into parts and summing those of them on which the points of the domain are situated, we obtain the measure of the domain as the limit of such sums when the number of parts increases without bound and—consequently—when they decrease without bound.

This definition, which takes no account whatsoever of the nature of the domain, was the first to appear in science; it was adopted successively, sometimes with slight variations, by Riemann, Stolz, Harnack (for two-dimensional domains), Peano, Jordan, and Lebesgue; the last two authors, not limiting themselves to the preceding definition, also introduce the inner measure of a domain, defining it as the limit of the sums of those intervals all of whose points belong to the given domain.

This first definition of measure, which we shall call Riemannian, belongs to those analysts whom the theory of point domains interested little in itself; having their own problems outside this theory, they made incidental use of its services and did not dwell especially long, except for Harnack and partly Jordan, on what might yield a more natural definition of measure.

A second group of investigators constructs around each point of the domain intervals enclosing it and takes the sums of their mutually non-overlapping parts; the measure of the domain is then defined as the limit of this sum, when each interval decreases without bound; such a definition is followed by Hankel, Dini, Harnack (in an early work), Cantor, and Lindelöf, who proposed a more flexible method of Cantor’s definition of measure.

The second definition of measure, which we shall call Hankelian, is considered more in accord with the nature of the domain than the first, but it also contains an element of arbitrariness: namely, the magnitudes of the intervals enclosing different points of the domain may stand in various relations to one another. That different choices of intervals may include, together with the points of the given domain, additional domains extremely different from one another, we see in the examples of Borel and Young.

The arbitrariness in the choice of the magnitudes of the intervals, as well as the arbitrariness in the choice of the system of divisions corresponding to the first definition of measure, gave occasion¹ to Schoenflies to express the opinion that “the definition

¹ Bericht, p. 87.

measures, like every mathematical definition, has a certain subjective character, and only the consequences flowing from it decide whether it has been chosen appropriately.” In the present case one can hardly agree with this, since the third definition of measure, as we shall see, is logically the only legitimate one.

An intermediate position between the first and second definitions is occupied by Lebesgue; then comes the unstable position of du-Bois-Reymond, who uses now the first, now the third definition¹.

Finally, the third definition takes as its basis free intervals, assuming—of course—that we are speaking of a row of scattered regions.

In determining measure, the domain of summation of the lengths of the free intervals has no influence on the individual lengths, and hence on their sum, whether the endpoints of these intervals are included in the region or not; therefore, in computing measure, all adjacent intervals merge into one, so that isolated points cannot have any influence on the measure; furthermore, the boundaries of the free intervals, which are limiting points of the region, likewise do not influence the measure, nor do common limiting points for two rows of intervals; such a possible removal of a countable row of points of the region, isolated and limiting, a removal not reflected in the magnitude of the measure, is equally unavoidable for all three definitions; in this respect the third definition is no exception.

Thus the question of defining the measure of an arbitrary region is at once reduced to the question of the measure of a perfect region or of a part of it; we touch upon this question below in Chapter III.

The first to whom such an idea of defining the measure of a region belongs is Smith, his regions² being of the second kind and having as their derivatives perfect regions.

Following Smith, the same definition was adopted by Volterra and de-Stefano for regions constructed according to the same pattern as Smith’s. Other investigators adjoining Smith, except for Harnack and partly Osgood, assume their regions to be perfect or closed; among these analysts are Veltmann, Borel, Schoenflies, and Young. Harnack generally makes no restrictive assumptions; as for Osgood, at the beginning he takes the region to be closed and then extends his definition.

Borel, as we saw above, does not give a definition of measure for a given set, but, on the contrary, singles out those sets to which his definition, based on continuous intervals, can be applied; but in the final account his measurable sets—according to Schoenflies’s theorems¹—are sets that are closed or complementary to closed sets.

Thus the third, Smith’s, definition branches into two currents: Smith’s and another, which may be called Veltmann’s; it is therefore desirable to determine in what measure they can be combined.

Among the general theorems concerning the measure of a set, one must note the theorems of Cantor, Borel, and Young. The theorems of the last-named author are of particular interest, since they make it possible to regard the first and second definitions of measure as consequences of Smith’s definition. These theorems put an end to discussions about the subjectivity of the definition, and so on, since the third definition is derived from the very nature of the set and is not connected with any restrictions on the question which one might have avoided.

Taking the third definition of measure as the starting point, we rid ourselves of the necessity, for linear sets, of proving that this definition does not depend on the sequence in which the intervals are chosen.

For frequently scattered sets the free intervals are absent; therefore here one must still examine how to extend the third definition of measure to them.

The first to propose a classification of sets according to whether the measure of the set is equal to or greater than zero was Harnack; we shall call the former—null (unausgedehnte), and the latter—full (ausgedehnte).

56. The question of the structure of a set attracted a greater number of workers and, to a larger extent, may be considered completed.

After, from the concepts of a one-to-one reciprocal correspondence of sets, of a limiting point (Bolzano), and of the scattering of sets over an interval (Hankel), Cantor established the concept of the derived set, thus laying the foundation of a new theory, its development proceeded quite rapidly.

Of the two questions that may be posed in this theory—1) to investigate the properties of a given set and 2) to construct a set under given conditions—the first problem at first attracted almost all attention.

The study of a given set led to the Cantor–Bendixson theorem concerning closed sets and to the concepts of adherence and coherence. The Cantor–Bendixson theorem may justly be called

the fundamental theorem in the theory of regions; Schoenflies and Young contributed to its geometric clarification. The fact that this theorem relates to closed regions does not diminish its general significance, since the study of non-closed regions must in one way or another rest upon the corresponding closed regions and—consequently—upon the fundamental theorem.

The appeal to free intervals (Dini) had as its consequence the classification of boundary points, interior and exterior points (Pincherle and Veltmann), which plays an important role in clarifying the structure of a region. Cantor’s classification of regions by genera and orders entailed a classification of limiting points (du Bois-Reymond and Cantor), which operated in the same direction. Thanks to all this, the structure of every infinite region is for us as clear as the simplest theorems of elementary geometry.

Besides regions of general form, especially in applications of the theory of regions, one often has to deal also with regions of special structure; these are: 1) regions consisting of various points that enter into the composition of an infinite sequence of regions, and 2) regions of points common to an infinite sequence of regions.

The first regions, from Ascoli and du Bois-Reymond, led to the classification of regions (Baire) into two categories, opening new perspectives; we see the same also with regard to the second regions, the investigations of which in the works of Borel and Young led to very interesting results.

With these and other regions we pass into the sphere of another problem: the determination of regions satisfying conditions prescribed in advance.

At first everything here was confined only to the construction of individual examples, which were to either prove or refute a known assertion, or else clarify a newly introduced concept. In the course of time a sufficient number of such examples accumulated, and it became desirable to introduce some order here; it was necessary to establish a general method, in the presence of which all the individual regions constructed earlier would turn out to be only particular cases.

The establishment of such a method is the subject of the second chapter of the present investigation; taking there three fundamental types of arrangement, we have tried to derive from them the whole variety of conceivable regions. Moreover, by introducing a notation for a known type, we can express the structure of a given region by a simple symbol and, having established, under given conditions, the corresponding type of arrangement, construct regions satisfying this type.

See §39. ↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩
See B, C 5°. ↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩
See, e.g., de la Vallée-Poussin, Cours d’analyse. ↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩
Proceedings, (2) 1, 1904, p. 246. ↩↩↩↩↩↩↩↩↩↩
ib., pp. 69–70. ↩↩↩↩
ib., p. 50. ↩↩
P. 83. ↩
Werke, 1876, p. 226. ↩

Submission history

[v1] 1907

	Page
1. Preliminary concepts and definitions	103
2. Rarefied sets	117
3. Basic and derivative types of arrangement	120
4. Complex types of arrangement	140
5. Mixed types of arrangement	176
6. Structure of an arbitrary set	201

Structure and Measure of Linear Point Sets: Preface. Chapter One

Full Text

Proceedings

of the Tomsk Technological Institute

of Emperor Nicholas II.

Structure and Measure of Linear Point Domains.

Contents

Chapter I

Chapter II

Chapter III

Chapter IV

Corrections

CHAPTER I.

Historical Sketch.

Submission history

Access Paper

Citation

Share

Related Papers

Feedback

Structure and Measure of Linear Point Sets: Preface. Chapter One