On the Semicontinuity in Product Spaces

Abstract

Let $X,Y$ be topological vector spaces or metric spaces, and let {$f:X\times Y \to \Re $} be a real function lower semicontinuous in the first variable and upper semicontinuous in the second one. It is proved that $f$ is globally measurable. Sierpinski (1925) has been raised this question in the case $X=Y=\Re $. This particular case was solved by Kempisty (1929). The actual result has applications in Calculus of Variations.