Suppose given a positive set-function $\mu$(F) in a base space R defined on a base class J of compact sets F. In this paper we obtain conditions under which $\mu$(F) determines a unique measure m(E) in R, finite on all compact subsets of R, and such that $\mu$(F) lies between the measure of F and that of the interior of F for every set F $\epsilon$ J. We assume $\mu$(F) to satisfy certain inequalities which are clearly necessary for our conclusions and show that if the class J is sufficiently big then every set-function $\mu$(F) satisfying these conditions does determine such a unique measure m(E). Different sufficient conditions on J 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.