Corollary

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

Integration, chapter IV, § 6. Convexity inequalities

— If the positive measure $\mu$ has total mass equal to 1 and if $f$ is integrable, then $\int f\, d\mu$ belongs to the closed convex envelope of $f(X)$ in $F$ .

It suffices to take for $g$ the constant function 1.

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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