Countable tightness in the spaces of regular probability measures

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 $wea{k}^{\ast }$ 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.