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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Extensions of measures and the von Neumann selection theorem


Author: Arthur Lubin
Journal: Proc. Amer. Math. Soc. 43 (1974), 118-122
MSC: Primary 28A10
DOI: https://doi.org/10.1090/S0002-9939-1974-0330393-X
MathSciNet review: 0330393
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Abstract: Let $ (X,{B_X})$ be a Blackwell space, where $ {B_X}$ is the $ \sigma $-algebra of Borel sets. Then if $ \sigma $ is a finite measure defined on a countably generated sub-$ \sigma $-algebra $ B \subset {B_X},\sigma $ can be extended to a Borel measure $ \tau $. Equivalently, if $ X$ and $ Y$ are Blackwell and $ f:X \to Y$ is Borel, and $ \mu $ is a Borel measure carried on $ f(X) \subset Y$, then there exists a Borel measure $ \tau $ on $ X$ with $ {\tau ^f} = \sigma $, where $ {\tau ^f}(E) = \tau ({f^{ - 1}}(E))$. We characterize $ \{ \tau \vert{\tau ^f} = \sigma \} $ if $ f$ is semischlicht.


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DOI: https://doi.org/10.1090/S0002-9939-1974-0330393-X
Article copyright: © Copyright 1974 American Mathematical Society