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Christensen measurability of polynomial functions and convex functions of higher orders. (English) Zbl 0616.28006
In this paper some results of P. Fischer and Z. Słodkowski [Proc. Am. Math. Soc. 79, 449-453 (1980; Zbl 0444.46010)] concerning Christensen measurable homomorphisms and Jensen convex functions are generalized to polynomial transformations and convex functions of higher orders. The notion of Haar zero sets in an Abelian Polish group was introduced by J. P. R. Christensen. Later the concept of Christensen measurability related to the family of Haar zero sets was considered by P. Fisher and Z. Słodkowski.
In the present paper it is shown that any Christensen measurable n-convex function defined on an open and convex subset of a real linear topological Polish space must be continuous. Moreover, some of S. Kurepa’s results describing certain properties of sets of positive Lebesgue measure in Euclidean spaces are extended to the case of non-zero Christensen measurable subsets of an arbitrary Abelian Polish group.

MSC:
28C10 Set functions and measures on topological groups or semigroups, Haar measures, invariant measures
28C15 Set functions and measures on topological spaces (regularity of measures, etc.)
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
39B99 Functional equations and inequalities
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