Lemma 17.15.41.E141137C118E Let $(\Gamma ^ i)_{i\in I}$ be a decreasing net in $\mathrm{TC}(X,\theta )_{>0}$ satisfying (17.15.38) and (17.15.39). Assume that $\omega $ is a closed smooth positive $(1,1)$-form on $X$. Then
\[ P_{\theta +\omega }\left[ \inf _{i\in I} \Gamma ^{i} \right]= \inf _{i\in I} P_{\theta +\omega }\left[ \Gamma ^{i} \right]. \]
Proof.
First observe that
\[ \left( P_{\theta +\omega }\left[ \inf _{i\in I} \Gamma ^{i} \right] \right)_{\max }= \left(\inf _{i\in I} P_{\theta +\omega }\left[ \Gamma ^{i} \right]\right)_{\max }. \]
Let $\tau \in \mathbb {R}$ be less than this common value. Then we need to show the following:
1D25957452B7
17.15.42
\begin{equation} P_{\theta +\omega }\left[ \inf _{i\in I} \Gamma ^ i_{\tau } \right]\sim _ P \inf _{i\in I} P_{\theta +\omega }\left[ \Gamma ^ i_{\tau } \right]. \end{equation}
Applying proposition 16.4.4 gives the mass convergence required in corollary 16.9.14. Hence $\Gamma ^ i_{\tau }\xrightarrow {d_ S}\inf _{j\in I} \Gamma ^ j_{\tau } $. Thanks to corollary 16.9.36 and corollary 16.9.14, we have
\[ P_{\theta +\omega }\left[ \Gamma ^ i_{\tau } \right]\xrightarrow {d_ S} P_{\theta +\omega }\left[ \inf _{j\in I} \Gamma ^ j_{\tau } \right],\quad P_{\theta +\omega }\left[ \Gamma ^ i_{\tau } \right]\xrightarrow {d_ S} \inf _{j\in I} P_{\theta +\omega }\left[ \Gamma ^ j_{\tau } \right]. \]
Hence, (17.15.42) follows from proposition 16.9.6.
$\square$
Pome
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