Affine upper envelope ($\overline{f}(x)=\inf \{a(x):a\in A(K),a\ge f\}$)

For a bounded real function on a compact convex set, this envelope is the infimum of its continuous affine majorants. It is finite, concave and upper semicontinuous, and dominates the function. For continuous functions it describes the maximal integral among probability measures with a specified barycenter. Its homogeneity, subadditivity and affine-translation identities give the supporting measure lemma for affine upper envelopes.

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.