Abstract

We prove that for any K-analytic subsets A,B of a locally compact group X if the product AB has empty interior (and is meager) in X, then one of the sets A or B can be covered by countably many closed nowhere dense subsets (of Haar measure zero) in X. This implies that a K-analytic subset A of X can be covered by countably many closed Haar-null sets if the set AAAA has an empty interior in X. It also implies that every non-open K-analytic subgroup of a locally compact group X can be covered by countably many closed Haar-null sets in X (for analytic subgroups of the real line this fact was proved by Laczkovich in 1998). Applying this result to the Kuczma–Ger classes, we prove that an additive function f:X→R on a locally compact topological group X is continuous if and only if f is upper bounded on some K-analytic subset A⊆X that cannot be covered by countably many closed Haar-null sets.

Highlights

  • By the classical Steinhaus Theorem [1], for any Lebesgue measurable subsets A of positive Lebesgue measure on the real line, the set A − A is a neighborhood of zero in R

  • In [2], Weil extended this result of Steinhaus to all locally compact topological groups proving that for any measurable subset A of positive Haar measure in a locally compact topological group X, the set AA−1 is a neighborhood of the identity in X

  • A Baire category analogue of the Steinhaus–Weil Theorem was obtained by Ostrowski [3], Piccard [4] and Pettis [5], who proved that for any nonmeager subset A with the Baire property in a Baire topological group X, the set AA−1 is a neighborhood of the identity

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Summary

Introduction

By the classical Steinhaus Theorem [1], for any Lebesgue measurable subsets A of positive Lebesgue measure on the real line, the set A − A is a neighborhood of zero in R. If a K-analytic subset A of a locally compact group X cannot be covered by countably many closed Haar-null sets in X, the product AA is not meager in X and AAAA has a nonempty interior in X. An additive function f : X → R on a locally compact group X is continuous if and only if sup f [A] < ∞ for some K-analytic set A ⊆ X, which cannot be covered by countably many closed Haar-null sets in X. Any discontinuous additive function f : Rω → R with f [L] = {0} witnesses that the linear space L does not belong to the Kuczma–Ger class CX Those examples show that Corollaries 1–5 cannot be generalized beyond the class of locally compact groups

Preliminaries
Findings
Proof of Theorem 1
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