Lemma 11.44.190E0F3A0EC129

Proof. By part (2) of corollary 15.13.3, $w_\lambda $ is the greatest admissible potential for $r=0$. If $m=\min _ Xr$ and $M=\max _ Xr$, then $w_\lambda -M/\lambda $ is admissible for $r$, and translating any potential admissible for $r$ by $m/\lambda $ makes it admissible for $r=0$. Hence

The mixed density inequality on $\Omega $ and the arithmetic–geometric mean inequality show that a convex combination of potentials admissible for $r_0$ and $r_1$ is admissible for the same convex combination of the forcing terms. Approximating the two suprema at $x$ proves concavity. Inclusion of the admissible classes proves monotonicity, and translation by constants proves the translation rule and the Lipschitz bound.

Now apply the Hahn–Banach construction in the proof of theorem 11.34.2 to $-\lambda \mathcal F_{\lambda ,x}$ at zero. This functional is finite, continuous, convex and increasing, and it satisfies $-\lambda \mathcal F_{\lambda ,x}(r+b)=-\lambda \mathcal F_{\lambda ,x}(r)+b$ for constants $b$. The same tests with nonnegative functions and constants show that its subgradient is a probability measure $\nu _{\lambda ,x}$. The resulting supporting inequality, combined with $s(x)\leq \mathcal F_{\lambda ,x}(r)$, gives (11.44.20); it is not necessary that the supremum be attained. $\square$