A characterization of the invertible measures

Abstract

Let G be a locally compact abelian group and M(G) its measure algebra. Two measures mu and lambda are said to be equivalent if there exists an invertible measure pi such that pi * mu = lambda. The main result of this note is the following: A measure mu is invertible iff vertical bar(mu) over cap vertical bar >= epsilon on (G) over cap for some epsilon > 0 and mu is equivalent to a measure lambda of the form lambda = a + theta, where a is an element of L-1(G) and theta is an element of M(G) is an idempotent measure.