Supporting measure lemma for affine upper envelopes ($\nu (f)=\mu (\overline{f}),\nu (g)\le \mu (\overline{g})\forall g\in C(K)$)

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.