11.44.7 Norming probabilities and integral boundsB51F7B7B7005

Proof. The formula for $\kappa $ follows from $N(a\mathrm{e}^{r/n})=c(\mathrm{e}^ rf)^{-1/n}$. The Hahn–Banach theorem gives a linear functional that norms $N$ at $a$. By (11.44.6) it is continuous on $L^\ell $, and hence is represented by some $\eta _0\in L^{\ell '}$. For $b\geq 0$, testing on $\operatorname {sgn}(\eta _0)b$ gives

Since $a\geq 0$ and $\int _ Xa\eta _0\, \mathrm{d}V=A=N(a)$, the same equality holds with $|\eta _0|$ in place of $\eta _0$. Monotonicity of $N$ extends the domination to real $b$, so $|\eta _0|\in \mathcal H_ a$. The defining conditions of $\mathcal H_ a$ are weakly closed and give $\| \eta \| _{\ell '}\leq C_ q$, so reflexivity proves weak compactness.

For nonzero nonnegative $b_0,b_1$, the weighted arithmetic–geometric mean inequality applied to $b_ j/N(b_ j)$, together with monotonicity and the triangle inequality, gives

Taking $b_ j=a\mathrm{e}^{r_ j/n}$ proves convexity of $\kappa $. Monotonicity and the translation rule follow from the corresponding properties of $N$, and together they imply the Lipschitz bound.

Every $\nu _\eta $ is a probability and satisfies

Jensen's inequality gives $\kappa (r)\geq \int _ Xr\, \mathrm{d}\nu _\eta $, which proves one inclusion in (11.44.10). The map $\eta \mapsto \nu _\eta $ is weakly continuous, since $ar\in L^\ell $ for $r\in C(X)$.

Conversely, let $\nu \in \mathcal P_ f$ and put $L(r)=\int _ Xr\, \mathrm{d}\nu $. Differentiating the curve $t\mapsto a\mathrm{e}^{tr/n}$ in $L^\ell $ at $t=0$ gives

Here the one-sided derivative $N'_+(a;\cdot )$ of the continuous convex function $N$ exists, and the error estimate follows from its Lipschitz bound. The inequality $L\leq \kappa $, applied to $tr$ and $-tr$, therefore gives

Thus $T(ar)=A L(r)$ defines a continuous linear functional on the space of products $ar$, $r\in C(X)$. These products are dense in

indeed, it suffices to approximate $b/a$ in $L^\ell (a^\ell \mathrm{d}V)$ by continuous functions. If $ar\geq 0$, then $ar=ar_+$ almost everywhere, and the first inequality in (11.44.12) gives $L(r-r_+)=0$, so $T(ar)\geq 0$. Approximating nonnegative elements of $S_ a$ by products $ar$ with $r\geq 0$ continuous shows that the continuous extension of $T$ to $S_ a$ is positive.

Represent this extension by some $\eta \geq 0$ in $L^{\ell '}$ vanishing on $\{ a=0\} $. Since $T(a)=A$, extending the second inequality in (11.44.12) to $S_ a$ and applying it to $b-a$ gives $T(b)\leq N(b)$ for $b\in S_ a$. For general $b\in L^\ell $,

Thus $\eta \in \mathcal H_ a$, and $\nu _\eta $ agrees with $\nu $ on all continuous functions. This proves the representation (11.44.10), and weak compactness of $\mathcal P_ f$ follows from that of $\mathcal H_ a$.

The bound (11.44.11) follows from Hölder's inequality and $\| \eta \| _{\ell '}\leq C_ q$. Under the last hypothesis, the norming inequality gives $\int _ X\mathrm{e}^{tz/n}\, \mathrm{d}\nu \leq 1$. Since $z$ is $\nu $-integrable by (11.44.11), Jensen's inequality implies $\int _ Xz\, \mathrm{d}\nu \leq 0$. $\square$