Measurable parametrizations and selections

Authors:
Douglas Cenzer and R. Daniel Mauldin

Journal:
Trans. Amer. Math. Soc. **245** (1978), 399-408

MSC:
Primary 28A20; Secondary 04A15, 54H05

DOI:
https://doi.org/10.1090/S0002-9947-1978-0511418-3

MathSciNet review:
511418

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Abstract | References | Similar Articles | Additional Information

Abstract: Let *W* be a Borel subset of (where ) such that, for each *x*, is uncountable. It is shown that there is a map, *g*, of onto *W* such that (1) for each *x*, is a Borel isomorphism of *I* onto and (2) both *g* and are -measurable maps. Here, if *X* is a topological space, is the smallest family containing the open subsets of *X* which is closed under operation (A) and complementation. Notice that is a subfamily of the universally or absolutely measurable subsets of *X*. This result answers a problem of A. H. Stone.

This result improves a theorem of Wesley and as a corollary a selection theorem is obtained which extends the measurable selection theorem of von Neumann.

We also show an analogous result holds if *W* is only assumed to be analytic.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1978-0511418-3

Article copyright:
© Copyright 1978
American Mathematical Society