Differentiation of Integrals

Abstract

No functions class for general measurable sets classes are known whose functions have the property of differentiability of integrals associated to such sets classes. In this paper,we give some subspaces of $L^s$ with $1<s<\infty$, whose functions are proven to have the differentiability of integrals associated to measurable sets classes in ${\bf R}^n $, this gives an answer to a question stated by Stein in his book Harmonic Analysis. We give also a example of some functions in these classes on ${\bf R}^2 $, which is continuous nowhere.