Premium
Pseudocompactness, Measurability, and Category in Compact Groups
Author(s) -
ITZKOWITZ GERALD L.
Publication year - 1992
Publication title -
annals of the new york academy of sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.712
H-Index - 248
eISSN - 1749-6632
pISSN - 0077-8923
DOI - 10.1111/j.1749-6632.1992.tb32254.x
Subject(s) - mathematics , haar measure , partition (number theory) , cardinality (data modeling) , metric space , locally compact space , combinatorics , group (periodic table) , discrete mathematics , sierpinski triangle , metrization theorem , compact space , pure mathematics , fractal , computer science , mathematical analysis , data mining , chemistry , organic chemistry , separable space
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 G δ with positive Haar measure in the group is proved. This proposition is an important tool used in the proof of the theorem concerning partitions.