Remark

Tag 09B3, permanent. Cite it as [Bourbaki, Tag 09B3].

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

— When $m$ is of the form $b\mu$ , where $b \in G$ and $\mu$ is a real measure on $T$ ,

I_{\Phi,m}(f) = \int \Phi(f(t), b)\,d\mu(t)

for every function $f \in \mathscr{L}_F^1(\mu)$ , because both members are continuous on this space and coincide when $f$ is a step function over the $|\mu|$ -integrable sets.

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

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Cites

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