# Reference for Choquet-like theorem

While reading a paper, I encountered the following statement:

Let $$K$$ be a convex compact subset of a locally convex topological vector space. If $$\mu \in P(K)$$ is a Radon probability measure on $$K$$, then there is a unique point $$x_\mu \in K$$ such that $$\int_K fd \mu = f(x_\mu)$$ for every continuous, affine, real-valued function $$f: K \to \mathbb{R}$$. Moreover, the map $$P(K) \to K: \mu \mapsto x_\mu$$ is a surjective affine map.

I consulted "Lectures on Choquet's theorem" but I couldn't find this version in the book. Does anybody know an appropriate reference/proof?

Edit: With an affine function $$f: K \to \mathbb{R}$$, I mean a function with the property

$$f\left(\sum_i \lambda_i k_i\right)= \sum_i \lambda_i f(k_i)$$ when $$0 \le \lambda_i \le 1$$ and $$\sum_i \lambda_i = 1$$, i.e. $$f$$ preserves convex combinations.

The point $$x_\mu$$ is the barycentre of the measure $$\mu$$. Its existence and uniqueness is Proposition 1.1 in "Lectures on Choquet's theorem" by Phelps.

• @R W But this proposition states something about $f \in E^*$ instead of affine functions! Oct 20 at 15:15
• What's the difference between affine and linear functions?
– R W
Oct 20 at 15:21
• @R W An affine map preserves convex combinations. A linear map is much stronger. Oct 20 at 15:23
• Not so much - think about it
– R W
Oct 20 at 15:28
• I presume Hahn - Banach would help
– R W
Oct 20 at 16:00