Difference properties for continuity and Riemann integrability on locally compact groups

Abstract

1. Introduction. Unless otherwise specified, (G, +) will always denote a (not necessarily abelian) locally compact topological group. Measure will denote a Haar measure on G; if G is compact, it is assumed that ,u(G) = 1. All functions considered are complex-valued. If f is a function on an open subset A of G, then Ahf and Vhf (hCG) denote the functions given by Ahf(x) =f(x+h) -f(x) and Vhf(x)=f(h+x)-f(x) on AC(A-h) and A(-h+A), respectively. A function r on G is said to be additive if r(x+y) =r(x) +r(y) for all x and y in G. A function f on an open subset A of G is said to be Riemann integrable if f is bounded on every compact subset of A, and if the set of points of discontinuity of f is of measure 0. The classes of continuous functions and Riemann integrable functions were among the many classes C of functions on the real line which, by results