Lemma 37.76.4. 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 37.41.5 to get an étale neighbourhood (U, u) \to (Y, y) and a decomposition
U \times _ Y X = W \amalg \ \coprod \nolimits _{i = 1, \ldots , n} \ \coprod \nolimits _{j = 1, \ldots , m_ i} V_{i, j}
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
\textstyle {\int }_{U \times _ Y X \to U} w|_{U \times _ Y X} = \textstyle {\int }_{W \to U} w|_ W + \sum \textstyle {\int }_{V_{i, j} \to U} w|_{V_{i, j}} = \textstyle {\int }_{W \to U} w|_ W + (\int _ f w)(y)
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 37.75.1). \square
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