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 ∈ℕ.

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, y∈X is continuous.

Theorem.Let G: X × X → ℝ be a mapping. If there is a subset M of X of positive finite Haar measure such that for each y∈X 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, y∈X 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.