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

 

Borel parametrizations


Author: R. Daniel Mauldin
Journal: Trans. Amer. Math. Soc. 250 (1979), 223-234
MSC: Primary 54H05; Secondary 04A15, 28A05
MathSciNet review: 530052
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Abstract: Let X and Y be uncountable Polish spaces and B a Borel subset oi $ X\, \times \,Y$ such that for each x, $ {B_x}$ is uncountable. A Borel parametrization of B is a Borel isomorphism, g, of $ X\, \times \,E$ onto B where E is a Borel subset of Y such that for each x, $ g\left( {x,\, \cdot } \right)$ maps E onto $ {B_x}\, = \,\left\{ {y:\,\left( {x,\,y} \right)\, \in \,B} \right\}$. It is shown that B has a Borel parametrization if and only if B contains a Borel set M such that for each x, $ {M_x}$ is a nonempty compact perfect set, or, equivalently, there is an atomless conditional probability distribution, $ \mu $, so that for each x, $ \mu \left( {x,\,{B_x}} \right)\, > \, 0$. It is also shown that if Y is dense-in-itself and $ {B_x}$ is not meager, for each x, then B has a Borel parametrization.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1979-0530052-3
PII: S 0002-9947(1979)0530052-3
Article copyright: © Copyright 1979 American Mathematical Society