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Proceedings of the American Mathematical Society

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Christensen zero sets and measurable convex functions

Authors: Pal Fischer and Zbigniew Słodkowski
Journal: Proc. Amer. Math. Soc. 79 (1980), 449-453
MSC: Primary 28C10; Secondary 39C05, 46A99
MathSciNet review: 567990
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Abstract: A notion of measurability in abelian Polish groups related to Christensen's Haar zero set is studied. It is shown that a measurable homomorphism or a measurable Jensen convex function defined on a real linear Polish space is continuous.

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

  • [1] G. Choquet, Lectures on analysis, vol. 1, Benjamin, New York, 1969.
  • [2] Jens Peter Reus Christensen, On sets of Haar measure zero in abelian Polish groups, Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972), 1972, pp. 255–260 (1973). MR 0326293,
  • [3] J. P. R. Christensen, Topology and Borel structure, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1974. Descriptive topology and set theory with applications to functional analysis and measure theory; North-Holland Mathematics Studies, Vol. 10. (Notas de Matemática, No. 51). MR 0348724
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  • [5] W. Sierpinski, Sur les fonctions convexes mesurables, Fund. Math. 1 (1920), 125-129.

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Keywords: Haar zero set, universally measurable set, convex functions
Article copyright: © Copyright 1980 American Mathematical Society