Nonstandard proofs of Eggleston like theorems

Details Nonstandard proofs of Eggleston like theorems Nonstandard proofs of Eggleston like theorems

2002-04-10

arxiv.org/abs/math/0204146v1

Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp. 353--357, Topology Atlas, Toronto, 2002

Mathematics

General Topology

5 pages

Scientific paper

We prove theorems of the following form: if $A\subseteq {\mathbb R}^2$ is a big set, then there exists a big set $P\subseteq {\mathbb R}$ and a perfect set $Q\subseteq {\mathbb R}$ such that $P\times Q\subseteq A$. 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 the proof given by J. Cichon. To prove these theorems we use Shoenfield's theorem about absoluteness for $\Sigma^1_2$-sentences.

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