The existence of universal invariant semiregular measures on groups

Abstract

A nonnegative, countably additive, extended real-valued measure is universal on a set X X iff it is defined on all subsets of X X , and is semiregular iff every set of positive measure contains a subset of positive finite measure. We prove that on every group of sufficiently large cardinality there exists a universal invariant semiregular measure vanishing on singletons. Thus we give complete solutions to the problems stated by Kannan and Raju [4] and Pelc [5].