Countable tightness in the spaces of regular probability measures
Volume 229 / 2015
Fundamenta Mathematicae 229 (2015), 159-169
MSC: Primary 46E15, 46E27, 28A33; Secondary 54C35.
DOI: 10.4064/fm229-2-4
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 Todorčević on measures on Rosenthal compacta.
