Nonstandard Proofs of Eggleston like Theorems
نویسنده
چکیده
We prove theorems of the following form: if A ⊆ R is a “big set”, then there exists a “big set” P ⊆ R and a perfect set Q ⊆ 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. Cichoń in [1]. To prove these theorems we use Shoenfield’s theorem about absoluteness for Σ12-sentences.
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