Lemma 17.5.11B5DC04E7729F

Proof. By lemma 17.5.10, $\varlimsup _ jq_ j=q$ outside a pluripolar set, hence $\nu $-almost everywhere. Thus $(q_ j-q)_+\to 0$ $\nu $-almost everywhere, and $\int _ X(q_ j-q)_+\, \mathrm d\nu \to 0$ by bounded convergence, since $0\leq (q_ j-q)_+\leq 2B$ and $\nu $ is finite. Since $|q_ j-q|=2(q_ j-q)_+-(q_ j-q)$, it remains to show that $\int _ Xq_ j\, \mathrm d\nu \to \int _ Xq\, \mathrm d\nu $. We have just seen that $\varlimsup _ j\int _ X(q_ j-q)\, \mathrm d\nu \leq 0$, so it suffices to prove that $L=\varliminf _ j\int _ Xq_ j\, \mathrm d\nu $ is at least $\int _ Xq\, \mathrm d\nu $.

Choose a subsequence $(q_{j_ k})$ with $\int _ Xq_{j_ k}\, \mathrm d\nu \to L$. For bounded Borel functions $y$ on $X$ put $\| y\| =(\int _ Xy^2\, \mathrm d\nu )^{1/2}$. For $l\geq 1$ let $C_ l$ be the set of finite convex combinations of the functions $q_{j_ k}$ with $k\geq l$, and let $d_ l=\inf \{ \| y\| :y\in C_ l\} $. Since $C_{l+1}\subset C_ l$ and $\| y\| \leq B\nu (X)^{1/2}$ for $y\in C_ l$, the numbers $d_ l$ increase to a finite limit. Choose $y_ l\in C_ l$ with $\| y_ l\| ^2\leq d_ l^2+1/l$. For $l\leq l'$ the function $\frac12(y_ l+y_{l'})$ belongs to the convex set $C_ l$, so its norm is at least $d_ l$. The parallelogram identity, obtained by integrating the pointwise identity $(a-b)^2+(a+b)^2=2a^2+2b^2$, therefore gives

and the right side tends to zero as $l\to \infty $, uniformly in $l'\geq l$.

By the Cauchy–Schwarz inequality, $\int _ X|y_ l-y_{l'}|\, \mathrm d\nu \leq \nu (X)^{1/2}\| y_ l-y_{l'}\| $, so we can choose $l_1<l_2<\cdots $ with $\int _ X|y_{l_{i+1}}-y_{l_ i}|\, \mathrm d\nu \leq 2^{-i}$. By monotone convergence the series $\sum _ i|y_{l_{i+1}}-y_{l_ i}|$ has finite integral, hence converges $\nu $-almost everywhere, and therefore $y_{l_ i}$ converges $\nu $-almost everywhere; let $h=\varlimsup _ iy_{l_ i}$, a Borel function with $|h|\leq B$. By bounded convergence, $\int _ Xy_{l_ i}\, \mathrm d\nu \to \int _ Xh\, \mathrm d\nu $. On the other hand, $\int _ Xy_ l\, \mathrm d\nu $ is a convex combination of the numbers $\int _ Xq_{j_ k}\, \mathrm d\nu $ with $k\geq l$, which tend to $L$, so $\int _ Xy_ l\, \mathrm d\nu \to L$. Hence $L=\int _ Xh\, \mathrm d\nu $.

It remains to show that $h=q$ $\nu $-almost everywhere. Each $y_{l_ i}$ is a convex combination of $\theta $-psh functions, hence $\theta $-psh, and $\| y_{l_ i}-q\| _{L^1(X)}\leq \sup _{k\geq l_ i}\| q_{j_ k}-q\| _{L^1(X)}$, which tends to zero. By lemma 17.5.10, applied to the sequence $(y_{l_ i})_ i$, the function $h=\varlimsup _ iy_{l_ i}$ equals $q$ outside a pluripolar set, hence $\nu $-almost everywhere. Thus $L=\int _ Xq\, \mathrm d\nu $, as required. $\square$