An application of a theorem of R. E. Zink
| Autor: | H. E. White |
|---|---|
| Rok vydání: | 1977 |
| Předmět: | |
| Zdroj: | Proceedings of the American Mathematical Society. 63:115-118 |
| ISSN: | 1088-6826 0002-9939 |
| DOI: | 10.1090/s0002-9939-1977-0432852-0 |
| Popis: | In ? 1 we discuss a measure theoretic analogue of Blumberg's theorem; in ?2 we discuss a topological analogue of the Saks-Sierpinski theorem. 1. In this section we discuss a measure theoretic analogue of Blumberg's theorem [7, 1.2]. Suppose (X, 5, I) is a totally finite measure space [2, p. 73], and 1* is the outer measure engendered by y. We consider the following statement. 1.1. For every real valued function f defined on X, there is a subset D of X such that ui*(D) = u(X) and fID is measurable (5 n D), where S n D = {S n D: S E )}. Actually 1.1 is equivalent to a special case of Blumberg's theorem. Let (X, c tic) denote the completion of (X, 5, Ly). By [3, p. 88], there is a topology %yc) on X such that (a) if U E %(Lc) and U 74 0, then U E oc and ,ic(U) > 0; and (b) if A E Sc, then there is U in '(yc) such that U c A and c (A U) = 0. It is easily verified that 1.1 holds for (X, 5, y) if and only if Blumberg's theorem holds for (X, C ( yc)). In [7, 2. 1] it is shown that 1.2. every subset of the closed unit interval I of cardinality R such that, if D is a subset of X for which f ID is measurable (5 n D), then u*(D) = 0. If 2"o = m, then clearly 1.2 holds; therefore, in this case, 1.1 is false for every separable, nonatomic measure space. However, there is a weaker statement-called Martin's axiom-which implies that 1.2 is true [5, ?4]. Presented to the Society May 17, 1976; received by the editors May 29, 1976. AMS (MOS) subject classifications (1970). Primary 28A20. |
| Databáze: | OpenAIRE |
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