Pseudocompactness, Measurability, and Category in Compact Groups

Abstract

It is shown that in the case where G is a compact topological group satisfying certain standard cardinality conditions, a theorem of Wilcox implies that it is possible to partition G into a collection of dense subsets, whose cardinality is Card(G), where each subset is pseudocompact. Thus each subset is of the second category in G, each subset is nonmeasurable, and each has Haar outermeasure one in G. This result simultaneously extends classical results of Ulam and Sierpinski on the partition of a perfect space into sets of category two and of Kakutani and Oxtoby on the partition of a compact metric group into nonmeasurable subsets of full Haar outermeasure. A useful proposition that in a compact group a subset has outermeasure one iff it meets every closed Gδ with positive Haar measure in the group is proved. This proposition is an important tool used in the proof of the theorem concerning partitions.