On the extreme points of moments sets

TL;DR

This paper characterizes the extreme points of measure sets with prescribed moments, providing necessary and sufficient conditions, and shows how such measures can be represented as combinations of Dirac measures, with applications to extremal problems.

Contribution

It offers new elementary conditions for identifying extreme measures with prescribed moments and demonstrates their representation as linear combinations of Dirac measures.

Findings

Conditions for extremality are expressed via atomic partitions.

Every extreme measure can be represented as a combination of Dirac measures.

Results apply to extremal problems over moment sets.

Abstract

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.