Corollary 1

Tag 08S5, permanent. Cite it as [Bourbaki, Tag 08S5].

Integration, chapter IV, § 4. Integrable functions and sets

— Let $a'$ be a continuous linear form on $F$ . If $f$ is an integrable function with values in $F$ , then the numerical function $\langle f, a' \rangle$ is integrable and

\int \langle f(x), a' \rangle \, d\mu(x) = \left\langle \int f(x) \, d\mu(x), a' \right\rangle.

We shall see in Ch. VI, § 1, Exers. 7, 11 and 12 that there can exist functions $f$ , with values in an infinite-dimensional Banach space $F$ , such that $\langle f, a' \rangle$ is integrable for every continuous linear form $a'$ on $F$ , without $f$ being integrable.

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

Cited by

Theorem 1, § 6

Cites

VI, § 1