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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 $\{ \tau |{\tau ^f} = \sigma \}$ if $f$ is semischlicht.


References [Enhancements On Off] (What's this?)

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Article copyright: © Copyright 1974 American Mathematical Society