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` **R**^{2} 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
\Sigma^{1}_{2}-sentences.

Mathematics Subject Classification. 03E15, 28A05.

Keywords. Lebesgue measure, perfect set.

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Topology Atlas.
Published April 2002.