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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Measurable parametrizations of sets in product spaces


Author: V. V. Srivatsa
Journal: Trans. Amer. Math. Soc. 270 (1982), 537-556
MSC: Primary 54H05; Secondary 04A15, 28A05
DOI: https://doi.org/10.1090/S0002-9947-1982-0645329-0
MathSciNet review: 645329
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Abstract: Various parametrization theorems are proved. In particular the following is shown: Let $ B$ be a Borel subset of $ I \times I$ (where $ I = [0,\,1]$) with uncountable vertical sections. Let $ \sum \dot \cup N$ be the discrete (topological) union of $ \sum $, the space of irrationals, and $ N$, the set of natural numbers with discrete topology. Then there is a map $ f:I \times (\sum \dot \cup N) \to I$ measurable with respect to the product of the analytic $ \sigma $-field on $ I$ (that is, the smallest $ \sigma $-field on $ I$ containing the analytic sets) and the Borel $ \sigma $-field on $ \sum \dot \cup N$ such that $ f(t,\,\, \cdot ):\,\sum \dot \cup N \to I$ is a one-one continuous map of $ \sum \dot \cup N$ onto $ \{ x:(t,\,x) \in B\} $ for each $ t \in T$. This answers a question of Cenzer and Mauldin.


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DOI: https://doi.org/10.1090/S0002-9947-1982-0645329-0
Keywords: Parametrizations, selections
Article copyright: © Copyright 1982 American Mathematical Society