Semiregular invariant measures on abelian groups

Abstract:

A nonnegative countably additive, extended real-valued measure is called semiregular if every set of positive measure contains a set of positive finite measure. V. Kannan and S. R. Raju [3] stated the problem of whether every invariant semiregular measure defined on all subsets of a group is necessarily a multiple of the counting measure. We prove that the negative answer is equivalent to the existence of a real-valued measurable cardinal. It is shown, moreover, that a counterexample can be found on every abelian group of real-valued measurable cardinality.