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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


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?)

  • [1] G. von Alexits, Über die Erweiterung einer Baireschen Funktion, Fund. Math. 15 (1930), 51-56.
  • [2] Donald L. Cohn, Measure theory, Birkhäuser, Boston, Mass., 1980. MR 578344 (81k:28001)
  • [3] H. J. Keisler and A. Tarski, From accessible to inaccessible cardinals. Results holding for all accessible cardinal numbers and the problem of their extension to inaccessible ones, Fund. Math. 53 (1963/1964), 225–308. MR 0166107 (29 #3385)
  • [4] E. L. Lehmann, Testing statistical hypotheses, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1959. MR 0107933 (21 #6654)
  • [5] E. Marczewski (Szpilrajn), The characteristic function of a sequence of sets and some of its applications, Fund. Math. 31 (1938), 207-223.
  • [6] E. Marczewski and R. Sikorski, Measures in non-separable metric spaces, Colloquium Math. 1 (1948), 133–139. MR 0025548 (10,23f)
  • [7] Dana Scott, Measurable cardinals and constructible sets, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 9 (1961), 521–524. MR 0143710 (26 #1263)
  • [8] W. Sierpiński, Sur l'extension des fonctions de Baire définies sur les ensembles linéaires quelconques, Fund. Math. 16 (1930), 31.
  • [9] S. Ulam, Zur Masstheorie in der allgemeinen Mengenlehre, Fund. Math. 16 (1930), 140-150.

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Additional Information

PII: S 0002-9939(1983)0684635-7
Keywords: Measurable space, extension property, separable space, standard space, measurable cardinal, two-valued measurable cardinal, separability character
Article copyright: © Copyright 1983 American Mathematical Society