Theorem 9.11.20CD69C52538E4

Proof. On the torus, polar coordinates turn the weighted monomial integral into a positive constant times

where $Q$ is the logarithmic image of the domain. The omitted toric divisors have measure zero on the regular locus. A basis of neighborhoods of $o$ is given by $|\chi ^{b_ j}|<\epsilon $ for every $j$, with $0<\epsilon <1$. Their logarithmic domains satisfy

for $b\in \operatorname {int}\sigma $ sufficiently far along an interior ray. Indeed, every $\langle b_ j,b\rangle $ is strictly positive, and the $b_ j$ generate the dual cone. The convex-cone lemma gives necessity and sufficiency at $o$, including divergence when a recession inequality is an equality.

For a compact subset of $X_\sigma $, the embedding coordinates have absolute value at most some $R>1$. Its logarithmic image is contained in $\{ x:\langle b_ j,x\rangle \leq \log R\} $, which is contained in $b_ R-\sigma $ for a sufficiently large $b_ R\in \operatorname {int}\sigma $. Under the same strict condition the integral over this translated cone is finite. This proves local integrability everywhere. The converse follows by testing at $o$, and the interior formulation follows from the lemma. $\square$