  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.
