Regularity properties of functional equations and inequalities

Summary By a well-known theorem of Lebesgue and Fréchet every measurable additive real function is continuous. This result was improved by Ostrowski who showed that a (Jensen-) convex real function must be continuous if it is bounded above on a set of positive Lebesgue measure. Recently, R. Trautner provided a short and elegant proof of the Lebesgue—Fréchet theorem based on a representation theorem for sequences on the real line.We consider here a locally compact topological groupX with some Haar measure. Then the following generalizes Trautner's theorem: Theorem.Let M be a measurable subset of X of positive finite Haar measure. Then there is a neighbourhood W of the identity e such that for each sequence (z _n )in W there is a subsequence (z _nk )and points y and x _k in M with z _nk =x _k ·y ^–1 for k . Using this theorem we obtain the following extensions of the theorems of Lebesgue and Fréchet and of Ostrowski. Theorem.Let R and T be topological spaces. Suppose that R has a countable base and that X is metrizable. If g: X R and H: R × X T are mappings where g is measurable on a set M of positive finite Haar measure and H is continuous in its first variable, then any solution f: X T of f(x · y) = H(g)(x), y) for x, yX is continuous. Theorem.Let G: X × X be a mapping. If there is a subset M of X of positive finite Haar measure such that for each yX the mapping x G(x, y) is bounded above on M, then any solution f: x of f(x · y) G(x, y) for x, yX is locally bounded above. We also prove category analogues of the above results and obtain similar results for general binary mappings in place of the group operation in the argument off.