Theorem 16.28.27865AACD2F96A

Proof. On the real Banach space $C(K)$, put $T(f)=F_{1,f}(a)$. By (16.28.25), $T$ is concave, since convex combinations of competitors are competitors. Comparison of obstacles and translation by constants show that $T$ is nondecreasing and satisfies

Fix $f_0\in C(K)$. The directional derivative of the convex Lipschitz function $-T$ at $f_0$ is finite and sublinear. By the Hahn–Banach theorem, there is a continuous linear functional dominated by it, and since difference quotients of a convex function are monotone, this functional supports $-T$ at $f_0$. Changing the sign, we obtain a continuous linear functional $\ell $ with

Testing $f=f_0+th$ with $h\geq 0$ and $t>0$, and using monotonicity, gives $\ell (h)\geq 0$. Testing with $f=f_0\pm t$ gives $\ell (1)=1$. By the Riesz representation theorem, $\ell (f)=\int _ K f\, \mathrm{d}\mu $ for a Borel probability measure $\mu $ on $K$. Set $c=T(f_0)-\int _ K f_0\, \mathrm{d}\mu $. The supporting inequality at $f=0$ gives $c\geq T(0)=V_ K^*(a)\geq 0$. Together with (16.28.25), it shows that $(\mu ,c)\in \mathcal J_ a(K)$ and

Conversely, applying (16.28.26) to $u=F_{1,f_0}$, which lies below $f_0$ quasi-everywhere on $K$, shows that $T(f_0)\leq \int _ Kf_0\, \mathrm{d}\mu +c$ for every pair in $\mathcal J_ a(K)$. This proves (16.28.28) for $\delta =1$, with the minimum attained. The identity $F_{\delta ,q}=\delta F_{1,q/\delta }$ gives the general case, and the zero obstacle gives the last assertion. $\square$