Topology Atlas Document # ppae-32


Nonstandard proofs of Eggleston like theorems

Szymon Zeberski

Proceedings of the Ninth Prague Topological Symposium (2001) pp. 353-357

We prove theorems of the following form: if A subset or equal R2 is a ``big set'', then there exists a ``big set'' P subset or equal R and a perfect set Q subset or equal R such that P×Q subset or equal A. We discuss cases where ``big set'' means: set of positive Lebesgue measure, set of full Lebesgue measure, Baire measurable set of second Baire category and comeagre set. In the first case (set of positive measure) we obtain the theorem due to Eggleston. In fact we give a simplified version of a proof given by J. Cicho\'n. To prove these theorems we use Shoenfield's theorem about absoluteness for \Sigma12-sentences.

Mathematics Subject Classification. 03E15, 28A05.
Keywords. Lebesgue measure, perfect set.

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Copyright © 2002 Charles University and Topology Atlas. Published April 2002.