Proposition 12

Tag 08LW, permanent. Cite it as [Bourbaki, Tag 08LW].

Integration, chapter III, § 1. Measures on a locally compact space

— If $\mu$ is a bounded measure on $X$ and if $g$ is a bounded continuous mapping of $X$ into $\mathbf{C}$ , then the measure $g \cdot \mu$ is bounded and $\| g \cdot \mu \| \leq \| g \| \cdot \| \mu \|$ .

For every function $f \in \mathscr{K}(X; \mathbf{C})$ ,

|\mu(fg)| \leq \| \mu \| \cdot \| fg \| \leq \| \mu \| \cdot \| g \| \cdot \| f \|.

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

Cited by

Definition 3, § 1

Cites

Nothing.