How big is the weak-* closure of the set of all (finite) convex combinations of Bernoulli measures among all invariant probability measures?

I mean, we are in the symbolic space $\{1,2,...,d\}^{\mathbb{N}}$ and I would like to know, how big the (weak-*) closure of the set $C$ is, where $$C = \{\sum_{i=1}^{n}\alpha_{i}\eta_{i}: n \in \mathbb{N}; \sum_{i=1}^{n}\alpha_{i}=1; \eta_{i} = Ber(p^{i}_{1},...,p^{i}_{d}) \}.$$

Of course, any reference towards this topic may be useful.

Thanks a lot for your attention

How big is the weak-* closure of the set of all (finite) convex combinations of Bernoulli measures among all invariant probability measures?

I mean, we are in the symbolic space $\{1,2,\ldots,d\}^{\mathbb{N}}$ and I would like to know, how big the (weak-*) closure of the set $C$ is, where $$C = \left\{\sum_{i=1}^n \alpha_i \eta_i: n \in \mathbb{N}; \sum_{i=1}^n \alpha_i=1; \eta_i = \mathrm{Ber}(p^i_1,\ldots,p^i_d) \right\}.$$

Of course, any reference towards this topic may be useful.

Thanks a lot for your attention