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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

References

Berberian, S. K. (1965), Measure and integration (Macmillan, New York).Google Scholar
Franklin, S. P. (1965), ‘Spaces in which sequences suffice’, Fund. Math. 57, 107–115.CrossRefGoogle Scholar
Gulden, S. L., Fleischman, W. M. and Weston, J. H. (1970), ‘Linearly ordered topological spaces’, Proc. Amer. Math. Soc. 24, 197–203.CrossRefGoogle Scholar
Halmos, P. R. (1950), Measure theory (Van Nostrand, New York).CrossRefGoogle Scholar
Johnson, R. A. (1966), ‘On product measures and Fubini's theorem in locally compact spaces’, Trans. Amer. Math. Soc. 123, 112–129.Google Scholar
Johnson, R. A. (1969), ‘Some types of Borel measures’, Proc. Amer. Math. Soc. 22, 94–99.CrossRefGoogle Scholar
Kelley, J. L. (1955), General topology (Van Nostrand, New York).Google Scholar