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Abstract

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.