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On a Paper of Maurice Sion

Published online by Cambridge University Press:  20 November 2018

Mark Mahowald
Mark Mahowald*
Affiliation:
Xavier University
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Let M0 be the set of measures μ on the real line such that open sets are μ*-immeasurable. While attempting to find out whether a set μ*-measurable for all μ in Mo is mapped into a similar set by a continuous function of bounded variation, Maurice Sion develops a theory for what he calls variational measure (4). As an application of the theory, he gets conditions on a function f and a set of measures M in order that f map a set, which is μ*-measurable for all μ ∈ M, into a set of the same kind. In particular he proves for his class M2 (def. 2.5), the following theorem (4, § 8.11).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 11 , 1959 , pp. 409 - 415
Copyright
Copyright © Canadian Mathematical Society 1959

References

1. Gödel, K., The consistency of the axiom of choice and of the generalized continuum-hypothesis, Proc. Nat. Acad. Sci. U.S.A., U (1938), 556-7.Google Scholar
2. Sierpinski, W., Les ensembles projectifs et analytiques, Mémorial des Sciences Mathématiques, no. 112 (1950).Google Scholar
3. Sierpinski, W. and A. Zygmund, Sur une fonction qui est discontinue sur tout ensemble de puissance du continu, Fund. Math., J+ (1923), 316-18.Google Scholar
4. Sion, Maurice, Variational Measure, Trans. Amer. Math. Soc, 83 (1956), 205-21.Google Scholar