Countably tight dual ball with a nonseparable measure

Abstract

We construct a compact Hausdorff space $K$ such that the space $P(K)$ of Radon probabiblity measures on $K$ considered with the weak$^*$ topology (induced from the space of continuous functions $C(K)$) is countably tight which is a generalization of sequentiality (i.e., if a measure $\mu$ is in the closure of a set $M$, there is a countable $M'\subseteq M$ such that $\mu$ is in the closure of $M'$) but $K$ carries a Radon probability measure which has uncountable Maharam type (i.e., $L_1(\mu)$ is nonseparable). The construction uses (necessarily) an additional set-theoretic assumption (the $\diamondsuit$ principle) as it was already known, by a result of Fremlin, that it is consistent that such spaces do not exist. This should be compared with the result of Plebanek and Sobota who showed that countable tightness of $P(K\times K)$ implies that all Radon measures on $K$ have countable type. So, our example shows that the tightness of $P(K\times K)$ and of $P(K)\times P(K)$ can be different as well as $P(K)$ may have Corson property (C) while $P(K\times K)$ fails to have it answering a question of Pol. Our construction is also a relevant example in the general context of injective tensor products of Banach spaces complementing recent results of Avil\'es, Mart\'inez-Cervantes, Rodr\'iguez and Rueda Zoca.