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

Abstract

Abstract Let f: ℝ2 → ℝ be a function with upper semicontinuous and quasi-continuous vertical sections f x(t) = f(x, t), t, x ∈ ℝ. It is proved that if the horizontal sections f y(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].