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 ] .