Theorem 13.15.138B2A9680933C

Proof. Choose $T_0$ such that every tail $\mu _{u_ j}(\{ u_ j<-T\} )$ with $T\geq T_0$ is at most one. By lemma 13.15.3, $\mu _{u_ j}(X)\leq C(1+T_0)^{r_ n}+1$, so the tail condition already gives a uniform bound for the total masses.

With the cutoffs $\eta _ T$ of (13.15.10), the fixed-level argument in the proof of theorem 13.15.9, applied with $w\equiv 1$, gives

The masses of the measures on the left are bounded by the uniform bound for the total masses. Since $\eta _ T(-u)$ increases to one off the pole set as $T\to \infty $, monotone convergence proves $\mu _ u(X)<+\infty $.

For a continuous test function $f:X\to \mathbb R$, the error made by inserting $\eta _ T(-u_ j)$ is at most $\| f\| _\infty \mu _{u_ j}(\{ u_ j<-T\} )$, which is small for large $T$, uniformly in $j$. For the limit potential, the corresponding error tends to zero because $\mu _ u$ is finite and does not charge the pole set. Let first $j\to \infty $ with $T$ fixed, and then $T\to \infty $. $\square$