Maybe it is a good to note that a similar question:
*What is the weak-$^*$ closure of the set*
$$
D=\{\mu\in\mathcal{M}_\sigma: \mu\text{ is isomorphic to some }\nu\in C\}?
$$
has a dramatically different answer: **$D$ is dense in $\mathcal{M}_\sigma$**. This is a corollary to a result of Jean-Paul Thouvenot and Benjy Weiss (still unpublished as far as I know). Weiss announced it during a talk in Prague last year and mentioned that Dan Rudolph also knew that.