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Measurability of cross section measure of a product Borel set

Part of: Classical measure theory

Published online by Cambridge University Press:  09 April 2009

Roy A. Johnson
Roy A. Johnson
Affiliation:
Department of Mathematics Washington State UniversityPullman, Washington 99164, U.S.A.
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Abstract

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Suppose μ and ν are Borel measures on locally compact spaces X and Y, respectively. A product measure λ can be defined on the Borel sets of X x Y by the formula λ(M) = ∫ν(Mx) dμ, provided that vertical cross section measure ν(Mx) is a measurable function in x. Conditions are summarized for ν(Mx) to be measurable as a function in x, and examples are given in which the function ν(Mx) is not measurable. It is shown that a dense, countably compact set fails to be a Borel set if it contains no nonempty zero set.

MSC classification

Secondary: 28A20: Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence 28A35: Measures and integrals in product spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 28 , Issue 3 , November 1979 , pp. 346 - 352
Copyright
Copyright © Australian Mathematical Society 1979

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