I have a set of probability measures $\mathcal{I}$ on $\{0,1\}^{\mathbb{Z}^d}$ and a family $\mathcal{F} = \{\mu_\alpha: \alpha \in [0,1]\} \subset \mathcal{I}$. The set $\mathcal{I}$ is compact and convex. I proved that for every $\mu \in \mathcal{I}$ there exists a distribution $\gamma$ in $[0,1]$ such that $$\mu = \int^1_0 \mu_\alpha \gamma(d\alpha).$$ If $\nu$ is an extremal measure of $\mathcal{I}$, I want to prove that $\nu \in \mathcal{F}$.
This integral is related to the notion of mixture (Generalized notions of mixture). If I would have a finite mixture (finite convex sum) it is straightforward but in this case, I have an "infinite" convex sum. Approximation to finite measures didn't work. Any help is really appreciated.