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Proceedings of the American Mathematical Society

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The extension of measurable functions

Author: R. M. Shortt
Journal: Proc. Amer. Math. Soc. 87 (1983), 444-446
MSC: Primary 28A05; Secondary 28A20
MathSciNet review: 684635
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Abstract: Say a measurable space $ (Y,\mathcal{B})$ has the extension property (resp. the extension property in the restricted sense) if for every measurable space $ (X,\mathcal{S})$ and every subset $ A$ of $ X$ (resp. subset $ A$ of $ X$ with $ X\backslash A$ singleton), each function $ f:A \to Y$ measurable for $ \mathcal{S}(A) = \{ B \cap A:B \in \mathcal{S}\} $ may be extended to a measurable function $ g:X \to Y$. A countably generated and separated $ (Y,\mathcal{B})$ has the extension property if and only if it is a standard space, i.e. it is isomorphic to a Borel subset of the real line. The discrete space $ (Y,{2^Y})$ has the extension property in the restricted sense if and only if the cardinality of $ Y$ is not two-valued measurable.

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

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Keywords: Measurable space, extension property, separable space, standard space, measurable cardinal, two-valued measurable cardinal, separability character
Article copyright: © Copyright 1983 American Mathematical Society