= Supporting measure lemma for affine upper envelopes
{title2=$\nu(f)=\mu(\overline f),\quad\nu(g)\le\mu(\overline g)\ \forall g\in C(K)$}
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>.
Back to article page
