Extensions of Haar Measure for Compact Connected Abelian Groups

Abstract

We outline in this paper generalizations of some theorems of Hulanicki on the existence of dense subsets of small cardinality in product measure spaces and in compact groups. We then apply a special case of these results to show the existence of the Kakutani-Oxtoby measure for the case of compact connected Abelian topological groups. A more detailed paper will appear later on. DEFINITION. Let Cfc, (B be collections of nonvoid sets of a space X. Then Cfc is a weak base for (B if and only if given J3£(B there is an ,4 G G such that ACB. If A is a set then | A denotes the cardinal of A ; n will always denote an infinite cardinal. The following theorem generalizes Hulanicki [7, Theorem l ] .