Jensen convex functions bounded above on nonzero Christensen measurable sets
Abstract
We prove that every Jensen convex function mapping a real linear Polish space into ℝ bounded above on a nonzero Christensen measurable set is convex.
Keywords
Christensen measurability; Jensen convex function
References
2. Christensen J.P.R., 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), Israel J. Math. 13 (1972), 255–260.
3. Christensen J.P.R., Topology and Borel structure. Descriptive topology and set theory with applications to functional analysis and measure theory. North-Holland Mathematics Studies, Vol. 10. (Notas de Matemática, No. 51). North-Holland Publishing Co., Amsterdam–London; American Elsevier Publishing Co., Inc., New York, 1974.
4. Fischer P., Słodkowski Z., Christensen zero sets and measurable convex functions, Proc. Amer. Math. Soc. 79 (1980), no. 3, 449–453.
5. Kuczma M., An Introduction to the Theory of Functional Equations and Inequalities. Cauchy’s Equation and Jensen’s Inequality. Second edition, Birkhäuser Verlag AG, Basel–Boston–Berlin, 2009.
6. Report of Meeting, The Twenty–first International Symposium on Functional Equations, August 6 – August 13, 1983, Konolfingen, Switzerland, Aequationes Math. 26 (1984), 225–294.
Katedra Matematyki, Politechnika Rzeszowska im. Ignacego Łukasiewicza Poland
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