Sur les fonctions de deux variables dont les coupes sont des dérivées

Abstract:

This paper concerns relationships between the measurability of a function $f:{I^2} \to R$ (when $I = \langle 0,1\rangle$ and $R$ is the set of all real numbers) and its cross sections ${f_{{x_0}}}(y) = f({x_0},y)$ and ${f^{{y_0}}}(x) = f(x,{y_0})$. A function $g:I \to R$ is said to have property $({\mathbf {K}})$ if for each measurable set $A \subset I$ of positive measure the function $g$ is ponctuellement-discontinue (i.e., the set of continuities is dense) on the closure of the set of all density points of $A$. The main result is: If a function $f:{I^2} \to R$ is bounded and each ${f_x}$ has property $({\mathbf {K}})$ and each ${f^y}$ is a derivative, then $f$ is Lebesgue measurable.