On the existence and uniqueness of certain measures

Abstract

Suppose given a positive set-function μ ( F ) in a base space R defined on a base class F of compact sets F . In this paper we obtain conditions under which μ ( F ) determines a unique measure m ( E ) in R , finite on all compact subsets of R , and such that μ ( F ) lies between the measure of F and that of the interior of F for every set F ∈ F . We assume μ ( F ) to satisfy certain inequalities which are clearly necessary for our conclusions and show that if the class F is sufficiently big then every set-function μ ( F ) satisfying these conditions does determine such a unique measure m ( E ). Different sufficient conditions on F are given according as the sets F in ( a ) are convex polytopes, or have analytic boundaries, ( b ) have sectionally analytic boundaries, or ( c ) are general compact sets, and it is shown by examples that these conditions cannot be relaxed too much. Thus the conclusions under ( a ) no longer hold in the plane if we assume that the sets are starlike polygons or convex sets with sectionally analytic boundaries. Nor is it possible to replace the sets under ( b ) by closed Jordan domains.