Borel sets with σ-compact sections for nonseparable spaces

We prove that every (extended) Borel subset E of X 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 is an element of Y : (x, y) is an element of E}, x is an element of X, are sigma-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.