We give several refinements of known theorems on Borel uniformizations of sets with “large sections”. In particular, we show that a set B ⊂ [0,1] × [0,1] which belongs to $\Sigma {⁰}_{\alpha }$, α ≥ 2, and which has all “vertical” sections of positive Lebesgue measure, has a $\Pi {⁰}_{\alpha }$ uniformization which is the graph of a $\Sigma {⁰}_{\alpha }$-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.

We prove that every (extended) Borel subset E of X × Y, where X is complete metric and Y is Polish, can be covered by countably many extended Borel sets with compact sections if the sections $E_x=y\in Y:(x,y)\in E$, x ∈ X, are σ-compact. This is a nonseparable version of a theorem of Saint Raymond. As a by-product, we get a proof of Saint Raymond’s result which does not use transfinite induction.