Proposition 13.5.8A93A2F4C95C2

Proof. For $n=1$ the conclusion is the hypothesis, so let $n\geq 2$. Work on a relatively compact coordinate domain on which $T_ i=\mathrm{dd}^{\mathrm{c}}a_ i$ with plurisubharmonic potentials $a_ i$ bounded above, and put

Since $G_ j$ is contained in the level-$j$ set $\{ a_ i>-j\} $ of the tuple $(a_ i,\ldots ,a_ i)$, lemma 13.2.2 shows that the bounded measure $(\mathrm{dd}^{\mathrm{c}}a_{i,j})^ n$ agrees with $\langle T_ i^ n\rangle $ on $G_ j$. By positivity, therefore, on the whole coordinate domain

Now theorem 7.12.1, applied to the densities $\mathbf1_{G_ j}f_ i$, gives $\mathrm{dd}^{\mathrm{c}}a_{1,j}\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}a_{n,j}\geq \mathbf1_{G_ j}\prod _ if_ i^{1/n}\nu $. Since $G_ j$ is the level-$j$ set of the tuple $(a_1,\ldots ,a_ n)$, the left side agrees on $G_ j$ with the non-pluripolar product of the $T_ i$, by lemma 13.2.2. Restricting to $G_ j$ and using positivity (lemma 13.2.1), we obtain

The complement of $\bigcup _ jG_ j$ is the union $P$ of the pole sets of the $a_ i$. Every $\langle T_ i^ n\rangle $ gives zero mass to the whole pluripolar set $P$, not only to the pole set of $a_ i$, so the inequality $\langle T_ i^ n\rangle \geq f_ i\nu $ forces $f_ i=0$ $\nu $-almost everywhere on $P$, for every $i$. Hence $\mathbf1_{G_ j}\prod _ if_ i^{1/n}$ increases to $\prod _ if_ i^{1/n}$ $\nu $-almost everywhere as $j\to \infty $, and monotone convergence on every Borel subset of the coordinate domain removes the indicator. Since finitely many such coordinate domains cover $X$, and an inequality of measures that holds on each set of a finite open cover holds on $X$ (decompose a Borel set into Borel pieces, each contained in one set of the cover), the asserted inequality follows.

Finally, $\int _ Xf_ i\, \mathrm d\nu \leq \int _ X\langle T_ i^ n\rangle <\infty $. To integrate the product, we use the inequality between weighted geometric and arithmetic means: for $y_1,\ldots ,y_ m\in [0,\infty ]$ and weights $\lambda _ i>0$ with $\sum _ i\lambda _ i=1$,

If all $y_ i$ are positive and finite, this is lemma 2.2.4 for the exponential function on the probability space $\{ 1,\ldots ,m\} $ with masses $\lambda _ i$, on which the integrand takes the values $\log y_ i$; otherwise the left side vanishes or the right side is infinite. With $\lambda _ i=1/n$, it gives $\int _ X\prod _ if_ i^{1/n}\, \mathrm d\nu \leq \frac1n\sum _ i\int _ Xf_ i\, \mathrm d\nu <\infty $. $\square$