Reports of the Academy of Sciences of the USSR

1957, Volume 114, No. 6

MATHEMATICS

Academician S. L. SOBOLEV

EXTENSIONS OF SPACES OF ABSTRACT FUNCTIONS CONNECTED WITH THE THEORY OF THE INTEGRAL

The theory of integration of abstract functions, studied by Bochner \((^1)\) and I. M. Gelfand \((^2)\), is conveniently constructed, following Bochner, by closing the integration operator

\[ \int_{\Omega} \varphi(P)\,dP, \tag{1} \]

defined on the set \(\mathfrak{M}\) of step functions \(\varphi(P)\) with values in the Banach space \(X\), i.e. for functions given by the equation

\[ \varphi(P)=\xi_i,\quad P\in E_i \quad (i=1,2,\ldots,N), \tag{2} \]

where \(\xi_i\) are certain elements of \(X\); \(E_i\) are pairwise disjoint sets inside \(\Omega\), Lebesgue measurable.

An abstract function \(\varphi(P)\) is called measurable if it serves as the limit of an almost everywhere convergent sequence of elements of \(\mathfrak{M}\) (and hence, also of an enumerable set of such sequences). In addition to almost-everywhere convergence, we shall simultaneously consider another type of convergence, connected with a certain norm \(\|\varphi\|_Y\) in the space of abstract functions, for which Bochner takes

\[ \|\varphi\|_B=\int_{\Omega}\|\varphi(P)\|_X\,dP. \tag{3} \]

This \(\|\varphi\|_Y\) must be chosen so that from the convergence of \(\|\varphi_k\|\) to zero for \(\mathfrak{M}\) it would follow that

\[ \lim \int_{\Omega}\varphi_k(P)\,dP=0. \tag{4} \]

In this case, for all sequences \(\varphi_k\) convergent in themselves in the sense of the norm \(\|\ \|_Y\), there exists, and moreover uniquely, the limit of the integral \(\int \varphi_k(P)\,dP\). A measurable function \(\varphi(P)\) will be called integrable in the sense of the norm \(\|\ \|_Y\), if for it there exists an almost everywhere convergent sequence \(\varphi_k(P)\) from \(\mathfrak{M}\), converging simultaneously in itself in this norm, i.e.

\[ \|\varphi_k(P)-\varphi_l(P)\|_Y<\varepsilon \quad \text{for } k,l>N(\varepsilon). \tag{5} \]

Besides the Bochner norm, one can indicate other, broader norms that make it possible to construct a broader definition of the integral. Let, for example:

\[ \|\varphi(\mathbf P)\|_{\Phi_1} = \sup \frac{\left\|\int \omega(\mathbf P)\varphi(\mathbf P)\,dP\right\|_X} {\|\omega(\mathbf P)\|_{L_\infty}}; \tag{6} \]

\[ \|\varphi(\mathbf P)\|_{\Phi_p} = \sup \frac{\left\|\int \omega(\mathbf P)\varphi(\mathbf P)\,dP\right\|_X} {\|\omega(\mathbf P)\|_{L_{p'}}}, \qquad p>1,\qquad \frac1p+\frac1{p'}=1, \tag{7} \]

where \(\omega(\mathbf P)\) ranges over the set of measurable real step functions of the point \(\mathbf P\), and \(\|\omega(\mathbf P)\|_{L_\infty}\) denotes the least upper bound of the values assumed by \(\omega(\mathbf P)\) on a set of positive measure. (For \(\|\ \|_{\Phi_1}\), in computing the sup it is enough to restrict oneself to such \(\omega(\mathbf P)\) as take two values \(\pm1\).)

The norm \(\|\ \|_{\Phi_p}\) is a natural generalization of the norm

\[ \|\varphi(\mathbf P)\|_{B_p} = \left[\int \|\varphi\|_X^p\,dP\right]^{1/p}, \tag{8} \]

which could have been considered instead of the Bochner norm. The norms \(\|\ \|_{\Phi_1}\) or \(\|\ \|_{\Phi_p}\) are broader than the Bochner norm. In the case when \(X\) is simply the real axis, \(\|\ \|_{\Phi_1}\) coincides with this norm; however, it is easy to give examples where this is no longer so.

Example 1. Let \(\varphi(x)\) be defined on the interval \(0\le x\le 1\) and take its values in \(l_2\), the space of real sequences with convergent sum of squares. Denote by \(i_k\) the element whose \(k\)-th component is equal to one and all the others are equal to zero. Define the function \(\varphi(x)\) by the formula \(\varphi(x)=2^k i_k/k\) on the interval \(2^{-k}<x\le 2^{-k+1}\). Then \(\varphi(x)\) is Bochner integrable.

Adjoining to \(\mathfrak M\) all abstract functions of a point \(\varphi(\mathbf P)\) that serve as limits, in the norm \(\|\ \|_{\Phi_p}\) and almost everywhere, of sequences \(\varphi_k(\mathbf P)\) does not turn this space into a complete one.

In order to obtain a space complete in the sense of \(\|\ \|_{\Phi_p}\), unlike the Bochner case, it is necessary to complete \(\mathfrak M\) by “ideal elements”—the limits of sequences converging in themselves in the sense of the norm \(\|\ \|_{\Phi_p}\). We shall show how this is to be done. Corresponding to each point function integrable in the sense of \(\|\ \|_{\Phi_p}\), we associate a certain additive abstract set function \(\varphi(E)\), with values in our Banach space \(X\), by the formula

\[ \varphi(E)=\int_E \varphi(\mathbf P)\,dP = \int_\Omega \eta_E(\mathbf P)\varphi(\mathbf P)\,dP, \tag{9} \]

where \(\eta_E(\mathbf P)\) is the characteristic function of the set \(E\). Instead of \(\varphi(\mathbf P)\), we shall study these set functions \(\varphi(E)\). In this way the integral of \(\varphi(\mathbf P)\) will correspond to \(\varphi(\Omega)\). From this point of view, the very formulation of the question of the integral as an operator acting from the set \(\mathfrak M\) into \(X\) loses its former meaning. Instead of studying extensions of such an operator, we shall study the internal properties of the new functional space we obtain.

Let \(\Phi_p\) denote the space of all additive abstract functions of sets \(\varphi(E)\), defined on all Lebesgue-measurable sets \(E\subset\Omega\), with norm

\[ \|\varphi(E)\|_{\Phi_p} = \sup \frac{\left\|\sum a_i\varphi(E_i)\right\|_X} {\|\omega(\mathbf P)\|_{L_{p'}}}, \qquad p\ge 1,\qquad \frac1p+\frac1{p'}=1, \]

\[ \omega(\mathbf P)=\alpha_i,\quad \mathbf P\in E. \tag{10} \]

Examples show that among the elements of \(\Phi_p\) there are some that cannot be constructed by formula (8).

Example 2. Let \(\Omega\) \((0 \le x \le 1)\); let \(X\) be \(l_2\); \(\psi_1(x)=i_1\); \(\psi_2(x)=i_2\), \(0 \le x \le 1/2\); \(\psi_2(x)=i_3\), \(1/2 < x \le 1\); \(\psi_3(x)=i_4\), \(0 \le x \le 1/4\); \(\psi_3(x)=i_5\), \(1/4 < x \le 2/4\); \(\psi_3(x)=i_6\), \(2/4 < x \le 3/4\); \(\psi_3(x)=i_7\), \(3/4 < x \le 1\), etc. Put

\[ \psi_s(E)=\int_E \psi_s(P)\,dP. \]

The series \(\sum_{s=1}^{\infty}\frac{2s^{1/2}}{s}\psi_s(E)=\varphi_0(E)\) converges in the norm \(\|\ \|_{\Phi_1}\) and defines a function \(\varphi_0(E)\) which is not the integral of any \(\varphi_0(P)\).

The set \(Y_\Omega\) of all measurable sets \(E\) such that \(E \subset \Omega\) may be regarded as a metric space by taking as the distance between two elements \(E_1\) and \(E_2\) the measure of their symmetric difference

\[ \rho(E_1,E_2)=m[(E_1\setminus E_1\cap E_2)\cup(E_2\setminus E_1\cap E_2)]. \tag{11} \]

We shall call a function \(\varphi(E)\) continuous in \(Y_\Omega\) if

\[ \|\varphi(E_1)-\varphi(E_2)\|_X<\varepsilon \quad \text{when } \rho(E_1,E_2)<\delta(\varepsilon). \tag{12} \]

For functions \(\varphi(E)\) that are integrals of point functions, this concept corresponds to absolute continuity. We introduce a more general concept, corresponding to the absolute continuity of integrals \(\int |\varphi|^p\,d\Omega\), by means of the norm \(\|\ \|_\Phi\).

Let \(\|\ \|_\Phi\) be some norm in the space \(\Phi\) of abstract set functions. To each \(\varphi(E)\) one may put in correspondence a new function \(\psi_\varphi(E_1)\) with values in \(\Phi\) by the formula

\[ \psi_\varphi(E_1)=\varphi(E\cap E_1). \tag{13} \]

We shall call a function \(\varphi(E)\) absolutely continuous in the norm \(\Phi\) if \(\|\psi_\varphi(E_1)\|_\Phi<\varepsilon\) as soon as \(mE_1<\delta(\varepsilon)\). We shall call the norm \(\|\ \|_\Phi\) monotone if \(\|\varphi(E)\|_\Phi \ge \|\varphi(E\cap E_0)\|\) for all \(E_0\). For a monotone norm, Theorem 1 holds.

Theorem 1. The function \(\psi_\varphi(E_1)\) corresponding to an absolutely continuous \(\varphi(E)\) will be continuous in the space \(Y\). Conversely, from the continuity of \(\psi_\varphi(E_1)\) follows the absolute continuity of \(\varphi(E)\).

Using the classical Weierstrass theorems of the theory of continuous functions and applying them to \(\psi_\varphi(E_1)\), we obtain Theorem 2.

Theorem 2. The limit of a sequence of functions \(\varphi_k(E)\), absolutely continuous in the norm \(\Phi\), converging in this norm, will be absolutely continuous.

The norms \(\|\ \|_{\Phi_p}\) will be monotone. Taking into account that the set \(\mathfrak M\) consisted of continuous functions, we obtain the following consequence.

The closure \(\Psi_p\) of the set \(\mathfrak M\) in \(\|\ \|_{\Phi_p}\) consists of functions absolutely continuous in \(\Phi_p\).

A well-known example shows that \(\Phi_p\) may also contain functions that are not absolutely continuous.

Example 3. Let \(\varphi_\delta(E)=1\) if \(0\in E\); \(\varphi_\delta(E)=0\) if \(0\notin E\). The function \(\varphi_\delta(E)\) will have bounded norm, but will not be absolutely continuous.

For sets \(E\), as consisting of vectors, one can define addition following Minkowski. Consider the norm of the difference \(\varphi(E+Q)-\varphi(E)\), where \(Q\) is a vector. If

\[ \|\varphi(E+Q)-\varphi(E)\|_\Phi<\varepsilon \quad \text{when } |Q|<\delta(\varepsilon), \tag{14} \]

then we shall say that the function \(\varphi(E)\) is continuous under translation. We shall call the norm \(\|\ \|_{\Phi}\) homogeneous if

\[ \|\varphi(E+Q)\|_{\Phi}=\|\Phi(E)\|_{\Phi}. \tag{15} \]

To each function \(\varphi(E)\) from \(\Phi\) one can put in correspondence an abstract function of the vector \(Q\), with values in \(\Phi\), by the formula

\[ Z_{\varphi}(Q)=\varphi(E+Q). \tag{16} \]

Theorem 3. If the function \(\varphi(E)\) is continuous under translation in the homogeneous norm \(\|\ \|_{\Phi}\), then \(Z_{\varphi}(Q)\) is continuous, and conversely.

In the same way as above, Theorem 4 is proved.

Theorem 4. A sequence \(\varphi_k(E)\) of elements continuous under translation and convergent in the homogeneous norm \(\Phi\) has as its limit a function continuous under translation.

It follows from this theorem that \(\Psi_p\) consists of functions continuous under translation. An example shows that in \(\Phi_p,\ p\geqslant 1\), there may exist absolutely continuous functions which are not continuous under translation.

Example 4. Define the functions \(\chi_s(x)\), with values in the \(m\)-space of bounded sequences, by the equalities \(\chi_1(x)=i_1,\ 0\leq x\leq 1/2;\ \chi_1(x)=-i_1,\ 1/2<x\leq 1;\ \chi_2(x)=i_2,\ 0\leq x\leq 1/4;\ \chi_2(x)=-i_2,\ 1/4<x\leq 2/4;\ \chi_2(x)=i_2,\ 2/4<x\leq 3/4;\ \chi_2(x)=-i_2,\ 3/4<x\leq 1\), and so on. Further, let

\[ \chi_s(E)=\int_E \chi_s(x)\,dx,\qquad \chi_0(E)=\sum_{s=1}^{\infty}\chi_s(E) \]

(the convergence is weak, i.e. for every \(E\), but not in norm!). For \(p=1\), \(\chi_0(E)\) will be absolutely continuous in \(\Phi_1\), but not continuous under translation.

A special role among functions of sets \(\varphi(E)\) is played by those which are representable in the form

\[ \varphi(E)=\int_E \varphi(P)\,dP, \tag{17} \]

where \(\varphi(P)\) are continuous abstract functions of the point \(P\). It is not difficult to define the integral of discontinuous abstract functions. The set of such functions \(\varphi(E)\) we shall denote by \(\mathfrak{R}\).

Theorem 5. Every function from \(\mathfrak{R}\) serves as the limit, in the sense of any \(\Phi_p\), of a sequence of elements from \(\mathfrak{R}\).

Theorem 6. The set of those elements of \(\Phi_p\) which possess absolute continuity and continuity under translation coincides with the closure of the set \(\mathfrak{R}\).

Thus Theorem 6 singles out a closed subspace serving as the closure of \(\mathfrak{R}\), and hence also \(\mathfrak{M}\). It follows from it that \(\Psi_p\) is completely characterized by two properties: absolute continuity and continuity under translation. Theorem 6 is proved by constructing mean functions for the abstract functions \(Z_{\varphi}(Q)\) generated by translation. These mean functions are continuous and give an approximation to \(\varphi(E)\) by means of elements of \(\mathfrak{R}\).

In the following note we shall show how embedding theorems for functional spaces, used in the theory of partial differential equations, can be transferred to abstract functions of sets.

Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
22 II 1957

REFERENCES CITED

1. S. Bochner, Fund. Mat., 20, 262 (1933).
2. I. Gelfand, Matem. sborn., 4(46), No. 2, 235 (1938).