Lemma 37.73.4. Let $f : X \to Y$ be a separated, locally quasi-finite morphism with finite fibres. Let $w : X \to \mathbf{Z}_{> 0}$ be a positive weighting of $f$. Then $\int _ f w$ is lower semi-continuous.

**Proof.**
Let $y \in Y$. Let $x_1, \ldots , x_ r \in X$ be the points lying over $y$. Apply Lemma 37.40.5 to get an étale neighbourhood $(U, u) \to (Y, y)$ and a decomposition

as in locus citatus. Observe that $(\int _ f w)(y) = \sum w(v_{i, j})$ where $w(v_{i, j}) = w(x_ i)$. Since $\int _{V_{i, j} \to U} w|_{V_{i, j}}$ is locally constant by definition, we may after shrinking $U$ assume these functions are constant with value $w(v_{i, j})$. We conclude that

This is $\geq (\int _ f w)(y)$ and we conclude because $U \to Y$ is open and formation of the integral commutes with base change (Lemma 37.72.1). $\square$

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