The Stacks project

Lemma 37.76.5. Let $f : X \to Y$ be a locally quasi-finite morphism with $X$ quasi-compact. Let $w : X \to \mathbf{Z}$ be a weighting of $f$. Then $\int _ f w$ attains its maximum.

Proof. It follows from Lemma 37.76.1 and Properties, Lemma 28.2.7 that $w$ only takes a finite number of values on $X$. It follows from Morphisms, Lemma 29.57.9 that $X \to Y$ has bounded geometric fibres. This shows that $\int _ f w$ is bounded. $\square$

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