2002/04/10 by Szymon Zeberski
Mathematics · #math.GN #msc:03E15 #msc:28A05
published as Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp. 353--357, Topology Atlas, Toronto, 2002 · 5 pages
arxiv created 2002/04/10 · arxiv updated 2009/11/30
We prove theorems of the following form: if A⊆ \mathbb R2 is a big set, then there exists a big set P⊆ \mathbb R and a perfect set Q⊆ \mathbb R such that P× Q⊆ 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 Σ12-sentences.