Countable tightness in the spaces of regular probability measures

TL;DR

This paper establishes that if the space of regular probability measures on the product of a compact space with itself has countable tightness, then the associated L1 spaces are separable, generalizing previous results under set-theoretic assumptions.

Contribution

It proves a new result linking measure space tightness to separability of L1 spaces, extending known theorems to broader contexts.

Findings

Countable tightness of P(K×K) implies separability of L1(μ) for all μ in P(K)

Generalizes Bourgain and Todorčević's theorem on measures on Rosenthal compacta

Provides a measure-theoretic condition related to topological tightness properties

Abstract

We prove that if $K$ is a compact space and the space $P(K\times K)$ of regular probability measures on $K\times K$ has countable tightness in its $weak^*$ topology, then $L_1(\mu)$ is separable for every $\mu\in P(K)$. It has been known that such a result is a consequence of Martin's axiom MA$(\omega_1)$. Our theorem has several consequences; in particular, it generalizes a theorem due to Bourgain and Todor\v{c}evi\'c on measures on Rosenthal compacta.