On a problem of Ulam
Abstract Generated abstract
The paper discusses a problem posed by Ulam concerning the existence, for every measure zero set, of a countably additive measure defined at least on its Borel subsets, normalized on the set, zero on points, and compatible with restriction to nested subsets. It notes that when a subset of the unit interval contains an uncountable Borel subset, the problem admits a simple solution by embedding a Cantor set and transferring Lebesgue measure to it. The paper then considers a strengthened version requiring invariance under ordinary congruence of subsets, and indicates that, using the previously introduced notion of alpha length, such a measure can be constructed for certain classes of sets.
Full Text
ON A PROBLEM OF ULAM
L. E. PORTNOV
(Presented in May 1966 to the mathematical section of the scientific-technical conference of AVTF, dedicated to the 70th anniversary of TPI)
In [1] the following problem was posed: does there exist, for every set \(A\) of measure \(0\) (say, on an interval), a countably additive measure \(m_A\) with respect to which at least all Borel subsets of \(A\) are measurable, and which has the properties:
1) \(m_A(A)=1;\qquad m_A(P)=0,\) where \(P\) is a point of \(A\),
2) for \(A \supset B \supset C\)
\[
m_A(C)=m_A(B)m_B(C).
\]
If the set \(A \subset [0,1]\) under consideration contains an uncountable Borel subset \(B\), then the problem has a simple solution satisfying conditions 1) and 2).
Indeed, by virtue of P. S. Aleksandrov’s theorem [2], there exists a Cantor set \(P \subset B\). On the set \(P\) one may induce Lebesgue measure from the interval \([0,1]\), using the known [3] one-to-one correspondence between the Cantor set without a countable set of one-sided points and the interval \([0,1]\), and therefore, putting for \(X \subset A\)
\[
m_A X = m_P (X \cap P),
\]
we obtain a measure \(m_A\) satisfying conditions 1) and 2).
We consider another problem, adding to conditions 1) and 2) condition 3): if \(B, C \subset A\) are congruent sets in the ordinary sense, then
\[
m_A B = m_A C.
\]
Using the \(\alpha\)-length considered in [4] and [5], a measure is constructed for certain classes of sets that satisfies conditions 1), 2), 3).
REFERENCES
- S. Ulam. Unsolved Mathematical Problems, 1964, p. 93.
- N. N. Luzin. Lectures on Analytic Sets and Their Applications, 1953.
- N. N. Luzin. Theory of Functions of a Real Variable, 1940.
- L. E. Portnov. Questions of Mathematics and Mechanics, TSU, vol. 179, 1966, pp. 66–78.
- L. E. Portnov. On the construction of oriented curves. Scientific-technical conference of young scientists and specialists. Abstracts of reports, 1966.