Borel classes of uniformizations of sets with large sections

Petr Holický¹

¹ Department of Mathematical Analysis Faculty of Mathematics and Physics Charles University Sokolovská 83 186 75 Praha 8, Czech Republic

Fundamenta Mathematicae, Tome 207 (2010) no. 2, pp. 145-160

Résumé

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.

Détail

DOI : 10.4064/fm207-2-3

Keywords: several refinements known theorems borel uniformizations sets large sections particular set subset times which belongs sigma alpha alpha which has vertical sections positive lebesgue measure has alpha uniformization which graph sigma alpha measurable mapping get similar result sets nonmeager sections corollary derive improvement srivastavas theorem uniformizations borel sets delta sections