Corollary

Tag 09MV, permanent. Cite it as [Bourbaki, Tag 09MV].

Integration, chapter IX, § 1. Premeasures and measures on a topological space

— Let $v$ and $w$ be two positive premeasures on $T$ , such that $v^\bullet(L) = w^\bullet(L)$ for every compact subset $L$ of $T$ ; then $v = w$ . In particular, the relation $v^\bullet = w^\bullet$ implies $v = w$ .

For, let $K$ be a compact set in $T$ ; for every compact set $L \subset K$ , one has the relation

w_K(L) = w_K^\bullet(L) = w^\bullet(L) = v^\bullet(L) = v_K^\bullet(L) = v_K(L)

by Prop. 2; therefore $w_K = v_K$ (Ch. IV, §4, No. 10, Cor. 3 of Prop. 19), and finally $w = v$ by the definition of premeasures.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Corollary 4, § 1
Lemma 1, § 2
Proposition 3, § 2
Proposition 4, § 2
Theorem 1, § 3
Theorem 1 (Prokhorov), § 4

Cites

IV, §4, No. 10, Cor. 3
Prop. 2