Proposition 2

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

Integration, IX, p. 24, § 2. Operations on measures

If f is a positive locally $\mu$ -integrable function, then, for every function $g \in \mathscr{F}_+(T)$ , one has the relation

(f \cdot \mu)^\bullet(g) = \mu^\bullet(fg).

Indeed, for every compact set K in T,

(f \cdot \mu)_K^\bullet(g_K) = (f_K \cdot \mu_K)^\bullet(g_K) = \mu_K^\bullet(f_K g_K) = \mu_K^\bullet((fg)_K),

on using the definition of $f \cdot \mu$ and Prop. 3 of Ch. V, §5, No. 3. Prop. 2 follows from this on passing to the supremum over K.

Now let $x \in T$ and let V be a neighborhood of x such that $\mu^\bullet(f \varphi_V) < +\infty$ (Def. 2); then $(f \cdot \mu)^\bullet(V) = \mu^\bullet(f \varphi_V) < +\infty$ , therefore $f \cdot \mu$ is a measure.

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

Cited by

Lemma 4, § 2
Proposition 12, § 2
Proposition 3, § 2
Proposition 4, § 2

Cites

Def. 2
Prop. 3
V, §5, No. 3