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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

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
MathSciNet review: 511418
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Abstract: Let W be a Borel subset of $ I \times I$ (where $ I = [0,1]$) such that, for each x, $ {W_x} = \{ y:\,(x,y) \in W\} $ is uncountable. It is shown that there is a map, g, of $ I \times I$ onto W such that (1) for each x, $ g(x, \cdot )$ is a Borel isomorphism of I onto $ {W_x}$ and (2) both g and $ {g^{ - 1}}$ are $ S(I \times I)$-measurable maps. Here, if X is a topological space, $ S(X)$ is the smallest family containing the open subsets of X which is closed under operation (A) and complementation. Notice that $ S(X)$ 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: http://dx.doi.org/10.1090/S0002-9947-1978-0511418-3
PII: S 0002-9947(1978)0511418-3
Article copyright: © Copyright 1978 American Mathematical Society