PomeProposition 20.1.45.9D4AD4B277B4 Suppose that $\mathbb {D}$ is a nef b-divisor over $X$. Then
\[ \left( \mathbb {D},\ldots ,\mathbb {D} \right)=\operatorname{vol}\mathbb {D}. \]
Proposition 20.1.459D4AD4B277B4
Proof. Using 20.1.44 and 20.1.7, we may easily reduce to the case where $\mathbb {D}$ is nef and big. In this case, take a non-divisorial closed positive $(1,1)$-current $T$ in $\mathbb {D}_ X$ such that $\mathbb {D}(T)=\mathbb {D}$. Then we need to show that
\[ \operatorname{vol}\mathbb {D}=\operatorname{vol}T, \]
which is proved in theorem 20.1.15. $\square$
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