Monolithic spaces of measures

For a compact space $K$ we consider the space $P(K)$, of probability regular Borel measures on $K$, equipped with the $wea{k}^{\ast }$ topology inherited from $C(K{)}^{\ast }$. We discuss possible characterizations of those compact spaces $K$ for which $P(K)$ is ${\mathrm{\aleph }}_0$-monolithic. The main result states that under $\mathrm{♢}$ there exists a nonseparable Corson compact space $K$ such that $P(K)$ is ${\mathrm{\aleph }}_0$-monolithic but $K$ supports a measure of uncountable type.