The Bishop-de-Leeuw theorem says the following:
Let $V$ be a topological vector space with $V^*$ separating and $C \subseteq V$ a non-empty, compact, convex subset of $V$. Let $E$ be the set of extreme points of $C$ and $X= \overline{E}$ (which is compact Hausdorff). For every $c \in C$, there exists a Radon probability measure $\mu_c$ on the compact Hausdorff space $X$ such that
$$\forall \omega \in V^*: \omega(c) = \int_X \omega(x) \mu_c(dx)$$
I'm trying real hard to understand why this is a useful theorem or what it is trying to say. My notes mention it after proving the Krein-Milman theorem but does not use it in the sequel.
Why is this an important theorem? What is the theorem trying to say? From a probability theory point of view, this says
$$\forall c \in C: \exists \mu_c: \forall \omega \in V^*: \omega(c) = \mathbb{E}_{\mu_c}(\omega)$$
I.e. $\omega(c)$ can be seen as the expected value of the functional $\omega$ on $X$ w.r.t. the probability measure $\mu_c$.
Any insight in the applications of this theorem or why it is useful will be appreciated.