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# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

## Borel classes of uniformizations of sets with large sections

### Tom 207 / 2010

Fundamenta Mathematicae 207 (2010), 145-160 MSC: Primary 54H05; Secondary 54C65, 54E50. DOI: 10.4064/fm207-2-3

#### Streszczenie

We give several refinements of known theorems on Borel uniformizations of sets with “large sections”. In particular, we show that a set $B\subset [0,1]\times [0,1]$ which belongs to ${\bf\Sigma}^0_{\alpha}$, $\alpha\ge 2$, and which has all “vertical” sections of positive Lebesgue measure, has a ${\bf\Pi}^0_{\alpha}$ uniformization which is the graph of a ${\bf\Sigma}^0_{\alpha}$-measurable mapping. We get a similar result for sets with nonmeager sections. As a corollary we derive an improvement of Srivastava's theorem on uniformizations for Borel sets with $G_{\delta}$ sections.

#### Autorzy

• Petr HolickýDepartment of Mathematical Analysis
Faculty of Mathematics and Physics
Charles University
Sokolovská 83
186 75 Praha 8, Czech Republic
e-mail

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