 Supporting measure lemma for affine upper envelopes

Let $\mu$ be a Borel probability measure on the compact convex set $K$, and fix $f\in C(K,\mathbb{R})$. the sublinear functional $p(g)=\mu (\overline{g})$ has a linear supporting functional taking the value $p(f)$ at a specified continuous $f$, by the Hahn-Banach theorem. It is positive and normalized because $p$ has these values on constants. The Riesz-Markov-Kakutani representation theorem gives the probability $\nu$. Testing affine functions and their negatives shows that $\mu$ and $\nu$ have the same barycenter.