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We prove that if $K$ is a compact space and the space $P(K\times K)$ ofregular 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 hasbeen known that such a result is a consequence of Martin's axiomMA$(\omega_1)$.  Our theorem has several consequences; in particular, it generalizes a theoremdue to Bourgain and Todor\v{c}evi\'c on measures on Rosenthal compacta.

Countable tightness in the spaces of regular probability measures