Let $X$ be a locally convex Hausdorff topological vector space and $S \subseteq X$ be compact. Define $\text{Prob}(S)$ to be the set of probability measures on $S$, $\text{conv}(S)$ to be the convex hull of $S$ and $\overline{\text{conv}}(S)$ to be its closure. Show that the map $\phi: \text{Prob}(S) \to X$ given by $\phi(\mu) = \int_S x \mathrm d\mu(x)$ satisfies $\phi(\text{Prob}(S)) = \overline{\text{conv}}(S)$. Conclude that the convex hull of a compact set is again compact.
Attempt:
For the $\phi(\text{Prob}(S)) \subseteq \overline{\text{conv}}(S)$ direction, my idea for a proof is to express any $\mu\in\text{Prob}(S)$ as a limit of a convex combination of Dirac measures, then pull the limit out of the differential and then the integral, but I am not sure if I am allowed to do this.
For the $\phi(\text{Prob}(S)) \supseteq \overline{\text{conv}}(S)$ direction, my idea for a proof is to after picking $$x = \lim_\alpha \sum_{j=1}^{m(\alpha)} \lambda_j(\alpha) s_j(\alpha) \in \overline{\text{conv}}(S),$$ use that $$\lim_\alpha \int_S x \mathrm d \left( \sum_{j=1}^{m(\alpha)} \lambda_j(\alpha) \delta_{s_j}(\alpha)\right) (x) = \int_S x \mathrm d\left(\lim_\alpha \sum_{j=1}^{m(\alpha)} \lambda_j(\alpha) \delta_{s_j}(\alpha)\right) (x)$$ and that $$\lim_\alpha \sum_{j=1}^{m(\alpha)} \lambda_j(\alpha) \delta_{s_j}(\alpha) \in \text{Prob}(S),$$ but I am not sure if I am allowed to do that.
I am thinking Krein-Milman or something of the sort might be useful here, but I am not sure how to apply it.
Any ideas are appreciated. Thanks!