Borel sets with countable sections for nonseparable spaces

Abstract

We prove that every (extended) Borel subset E E of X × Y X\times Y , where X X is a complete metric and Y Y is Polish, can be covered by countably many extended Borel graphs of mappings from X X to Y Y if the sections E x = { y ∈ Y : ( x , y ) ∈ E } E_x=\{y\in Y:(x,y)\in E\} , x ∈ X x\in X , are countable. This is a nonseparable version of a classical theorem of Luzin and Novikov.