Proposition 17.15.40.82DA4D86CC78 Let $(\Gamma ^ i)_{i\in I}$ be a decreasing net in $\mathrm{TC}(X,\theta )_{>0}$ satisfying (17.15.38) and (17.15.39). Then $\inf _{i\in I} \Gamma ^{i}\in \mathrm{TC}(X,\theta )_{>0}$.
Moreover, if $\Gamma ^ i\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$ for all $i\in I$, then so is $\inf _{i\in I} \Gamma ^{i}$.
Proof.
The first assertion is an immediate consequence of proposition 16.4.3, except for the positivity of the mass at $-\infty $. Choose $\tau _0$ as in (17.15.39). Applying proposition 16.4.4 gives
\[ \int _ X \left(\theta +\mathrm{dd}^{\mathrm{c}}\inf _{i\in I}\Gamma ^ i_{\tau _0}\right)^ n =\lim _{i\in I}\int _ X\left(\theta +\mathrm{dd}^{\mathrm{c}}\Gamma ^{i}_{\tau _0}\right)^ n>0. \]
Hence the value at $-\infty $ has positive mass by monotonicity. The last assertion follows from proposition 16.6.29.
$\square$
Pome
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