Corollary

Tag 09B8, permanent. Cite it as [Bourbaki, Tag 09B8].

Integration, chapter VI, § 2. Vectorial measures . "

— Let m be a bounded complex measure. Every function $f \in \mathscr{L}_\mathbf{F}^\infty(m)$ is then m-integrable, and $\left| \int f \, dm \right| \leq N_\infty(f)\|m\|$ .

For, $f$ is $m$ -measurable and, setting $\mu = |m|$ , we have

\int^* |f| \, d\mu \leq N_\infty(f)\|\mu\| = N_\infty(f)\|m\|,

therefore

f

|m|

-integrable (Ch. IV, §5, No. 6, Th. 5) and

\left| \int f \, dm \right| \leq \int |f| \, d\mu \leq N_\infty(f)\|m\|.

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

Cited by

Nothing.

Cites

IV, §5, No. 6, Th. 5