On the differentiability of arbitrary real-valued set functions. II.

Abstract:

Let f be a real-valued function defined and finite on sets from a family $\mathcal {F}$ of bounded measurable subsets of Euclidean n-space such that if $T \in \mathcal {F}$, the measure of T is equal to the measure of the closure of T. An earlier paper [Trans. Amer. Math. Soc. 145 (1969), 439-454] considered the questions of finiteness and boundedness of the upper and lower regular derivates of f and of the existence of a unique finite derivative. The present paper is an extension of the earlier paper and considers the summability of the derivates. Necessary and sufficient conditions are given for each of the upper and lower derivates to be summable on a measurable set of finite measure. A characterization of the integral of the upper derivate is given in terms of the sums of the values of the function over finite collections of mutually disjoint sets from the family.