On the measurability of functions with quasi - continuous and upper semi - continuous vertical sections

Abstract

Let f : R2 → R be a function with upper semicontinuous and quasi-continuous vertical sections fx(t) = f(x, t), t, x ∈ R. It is proved that if the horizontal sections fy(t) = f(t, y), y, t ∈ R, are of Baire class α (resp. Lebesgue measurable) [resp. with the Baire property] then f is of Baire class α + 2 (resp. Lebesgue measurable and sup-measurable) [resp. has Baire property].