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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.

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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.

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    $\begingroup$ What's the difference between affine and linear functions? $\endgroup$ Commented Oct 20, 2021 at 15:21
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    $\begingroup$ Not so much - think about it $\endgroup$ Commented Oct 20, 2021 at 15:28
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    $\begingroup$ Ah an affine function is a translation of a linear function. But then there is still the issue of domain. Can any affine function on $K$ be extended to an affine function on the locally convex space? $\endgroup$ Commented Oct 20, 2021 at 15:46
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    $\begingroup$ I presume Hahn - Banach would help $\endgroup$ Commented Oct 20, 2021 at 16:00
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    $\begingroup$ @R W Does Hahn-Banach allow us to extend affine functions on the compact subset $K$? $\endgroup$ Commented Oct 20, 2021 at 18:46

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