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.