The Stacks project

Proof. Let $x \in X$ with image $y \in Y$. Choose an étale neighbourhood $(V, v) \to (Y, y)$ and an open $U \subset X_ V$ such that $\pi : U \to V$ is finite and there is a unique point $u \in U$ mapping to $v$ with $\kappa (u)/\kappa (v)$ purely inseparable. See Lemma 37.41.3. Then $(\int _\pi w|_ U)(v) = w(u)$. It follows from Definition 37.75.2 that after replacing $V$ by a neighbourhood of $v$ we we have $w|_ U(u') \leq w|_ U(u) = w(x)$ for all $u' \in U$. Namely, $w|_ U(u')$ occurs as a summand in the expression for $(\int _\pi w|_ U)(\pi (u'))$. This proves the lemma because the étale morphism $U \to X$ is open. $\square$