Lemma 19.5.12.5F65C388181A Assume that $T,S\in \mathcal{Z}_+(X,\alpha )$ are two Kähler currents with analytic singularities and $T\preceq S$. Then
\[ \Delta _{Y_{\bullet }}(T)\subseteq \Delta _{Y_{\bullet }}(S)\subseteq \Delta _{Y_{\bullet }}(\alpha ). \]
Moreover,
B544FB9D12E6
19.5.13
\begin{equation} \operatorname{vol}\Delta _{Y_{\bullet }}(T)=\frac{1}{n!} \int _ X T^ n. \end{equation}
Proof.
We first show that
\[ \Delta _{Y_{\bullet }}(T)\subseteq \Delta _{Y_{\bullet }}(S). \]
Using 19.5.11, we may assume that $T$ and $S$ have log singularities along effective $\mathbb {Q}$-divisors $E$ and $F$ respectively. By assumption, $E\geq F$. Replacing $T$ and $S$ by $T-[F]$ and $S-[F]$ respectively, we may assume that $F=0$, cf. (19.5.9).
In this case, we need to show that
\[ \Delta _{Y_{\bullet }}(\alpha )\supseteq \Delta _{Y_{\bullet }}(\alpha -\{ E\} )+\nu _{Y_{\bullet }}([E]), \]
which is obvious.
Next we prove that
\[ \Delta _{Y_{\bullet }}(T)\subseteq \Delta _{Y_{\bullet }}(\alpha ). \]
By 19.5.11 and theorem 19.5.4 again, we may assume that $T$ has log singularities. We take $D$ and $R$ as in (19.5.8). We need to show that
\[ \Delta _{Y_{\bullet }}(\alpha -\{ D\} )+\nu _{Y_{\bullet }}([D])\subseteq \Delta _{Y_{\bullet }}(\alpha ), \]
which is again obvious.
Finally, (19.5.13) follows immediately from theorem 19.5.4.
$\square$
Pome
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