Let $X$ be a compact set.
Consider the set $H$ of non-negative probability measure $\eta$ defined on $X$ satisfying $$ \eta(dx) = \delta_\xi(dx),\quad \xi \in X. $$
How to understand, how to expect that $$ \overline{\text{conv}}(H) = \{\text{probability measure}\}? $$
Say $B$ is the set of all Borel probability measures on $X$. Then $B$ is convex, and $B$ is compact in the weak* topology (regarding measures as elements of $C(X)^*$.) So $B$ is the weak* closed convex hull of its extreme points, by Choquet's theorem. It's not hard to show that the extreme points of $B$ are exactly the measures in $H$.