Theorem 14.25.3FE27B72E2DB5

Proof. Under (14.25.1), theorem 11.1.7 gives $\int _ X\mathrm{MA}_\omega (a)=V$ for every bounded potential $a$; for $n=1$ this is Stokes' theorem.

Set $q=\max (a,b)$. On $\{ a<b\} $, plurifine locality gives $\mathrm{MA}_\omega (q)=\mathrm{MA}_\omega (b)$. On $\{ a\geq b\} $, one has $a=q$, and $a\leq q$ everywhere, so bounded ordered contact (lemma 5.28.3) gives

The total masses of $a$, $b$ and $q$ all equal $V$. Hence

which is (14.25.4).

Conversely, comparison for smooth potentials implies that all smooth potentials have the same mass: given a smooth pair $a,b$, add to $b$ a constant so large that $\{ a<b\} =X$, apply (14.25.4), and then interchange the roles of $a$ and $b$. Comparison with the zero potential shows that the common mass is $V$. By theorem 11.1.7, lemma 11.1.3, this implies (14.25.1) when $n\geq 2$; for $n=1$ that condition holds for degree reasons. $\square$