  Necessary and sufficient conditions for a measure to be an extreme point of
the set of measures (on an abstract measurable space) with prescribed
generalized moments are given, as well as an application to extremal problems
over such moment sets; these conditions are expressed in terms of atomic
partitions of the measurable space. It is also shown that every such extreme
measure can be adequately represented by a linear combination of k Dirac
probability measures with nonnegative coefficients, where k is the number of
restrictions on moments; moreover, when the measurable space has appropriate
topological properties, the phrase "can be adequately represented by" here can
be replaced simply by "is". The proofs are elementary and mainly
self-contained.
