A dichotomy for the convex spaces of probability measures

TL;DR

This paper establishes a dichotomy for convex spaces of probability measures, showing they either contain measures with small local character or measures of large Maharam type, linking measure properties with Banach space characteristics.

Contribution

It introduces a new dichotomy for convex spaces of probability Radon measures, connecting measure-theoretic properties with Banach space theory.

Findings

Spaces contain measures with small local character or large Maharam type

Links between Radon measures on compact spaces and Banach space properties

Provides a structural classification of convex measure spaces

Abstract

We show that every nonempty compact and convex space M of probability Radon measures either contains a measure which has `small' local character in M or else M contains a measure of `large' Maharam type. Such a dichotomy is related to several results on Radon measures on compact spaces and to some properties of Banach spaces of continuous functions.