Theorem 11.44.23F95A960C6FA

of theorem 11.44.2. By proposition 11.44.9 and the moment bound following it, $\mathcal P_ f$ has the stated properties, and lemma 11.44.15 gives precompactness of the centered potentials. It remains to identify every cluster potential.

Step 1: limits of the forcing probabilities. Fix $x\in \Omega $. For $r\in C(X)$, choose a critical potential $\psi _ r$ for $\mathrm{e}^ r f$ with supremum zero. By the definition of $\kappa $,

Since $\psi _ r\leq 0$, the potential $\psi _ r-\kappa (r)/\lambda $ is admissible in (11.44.18). Thus

For fixed $x$ and $r$, the values $w_\lambda (x)$ are uniformly bounded by lemma 11.44.15, and $\psi _ r(x)$ is finite. Hence every weak cluster point of $\nu _{\lambda ,x}$ as $\lambda \downarrow 0$ belongs to $\mathcal P_ f$.

Step 2: a pointwise comparison. Let $s$ satisfy (11.44.3). The function $-\lambda s$ is lower semicontinuous and bounded below on $X$, with the value $+\infty $ allowed, so metric inf-convolution and truncation give continuous functions $r_ j\uparrow -\lambda s$. Since $r_ j\leq -\lambda s$, the potential $s$ is admissible for every $r_ j$, and (11.44.20) bounds the integrals of the $r_ j$ from above. Monotone convergence, after subtracting a common lower bound, shows that $s$ is integrable against $\nu _{\lambda ,x}$ and gives

Step 3: cluster potentials belong to the class (11.44.3). Take a sequence $\lambda _ j\downarrow 0$ with $w_{\lambda _ j}\to z$ locally uniformly on $\Omega $ and in $L^1(X)$. The bounds (11.44.16) show that $z$ has minimal type, and $z$ is continuous on $\Omega $. Since $c\mathrm{e}^{\lambda _ j w_{\lambda _ j}}f\to cf$ in $L^ p$, by dominated convergence and the common bound $w_\lambda \leq K$, continuity of local bounded products gives $\mathrm{MA}_\theta (z)=cf\, \mathrm{d}V$ on $\Omega $, and then on $X$ because neither side charges $X\setminus \Omega $.

For each $\nu \in \mathcal P_ f$, the bound $|w_\lambda |\leq |\psi |+K$ has a $\nu $-integrable right-hand side, and $\nu $ charges only $\Omega $. Dominated convergence in $\int _ Xw_\lambda \, \mathrm{d}\nu \leq 0$ therefore gives $\int _ Xz\, \mathrm{d}\nu \leq 0$. Hence $z$ belongs to the class (11.44.3).

Step 4: $z$ is the greatest element of the class. For any $s$ in the class and fixed $x\in \Omega $, pass to a further subsequence with $\nu _{\lambda _ j,x}\rightharpoonup \nu $, so that $\nu \in \mathcal P_ f$ by Step 1. The potential $s$ is upper semicontinuous and bounded above on $X$, so

For an unbounded $s$, this follows from the portmanteau theorem applied to $\max (s,-M)$, followed by $M\to \infty $; the final integral is finite by (11.44.11). Passing to the limit in (11.44.22) gives $z(x)\geq s(x)$. This holds for every $x\in \Omega $, and hence on $X$ by lemma 5.24.3. Thus every cluster potential is the greatest element $w$ of the class, and precompactness gives convergence of the whole family.

Step 5: the normalizations. The maximum of $\int _ Xw\, \mathrm{d}\nu $ over $\mathcal P_ f$ is attained, because under (11.44.10) it is the maximum of the weakly continuous functional $\eta \mapsto A^{-1}\int _ Xaw\eta \, \mathrm{d}V$ on $\mathcal H_ a$; note that $aw\in L^\ell $. If the maximum were negative, a small positive constant translate of $w$ would still satisfy (11.44.3), contrary to greatestness. Thus it is zero.

Finally, $k_\lambda \to \sup _ Xw$. Indeed, from any subsequence choose a further one with $k_\lambda \to k$. Then $v_\lambda =w_\lambda -k_\lambda $ converges in $L^1$ to $w-k$, and theorem 5.25.2 gives $\sup _ X(w-k)=0$. Thus $k=\sup _ Xw$ for every cluster point of $(k_\lambda )$. Subtracting the constants proves the convergence of $u_\lambda -\sup _ Xu_\lambda $. If $\theta $ is Hermitian, we may take $\rho =0$ and $\Omega =X$, so that local uniform convergence is uniform convergence. $\square$