Definition 1

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

Integration, III, p. 33, § 3. Integrals of continuous vector-valued functions

For every function $f \in \widetilde{\mathscr{K}}(X;E)$ we call integral of $f$ with respect to $\mu$ , and denote by $\int f d\mu$ or $\int f(x) d\mu(x)$ , or $\int f \mu$ , or $\int f(x)\mu(x)$ , the element of ${E'}^*$ defined by

\left\langle \int f d\mu, z' \right\rangle = \int \langle f, z' \rangle d\mu \quad \text{for all } z' \in E'.

We note that even if $E$ is Hausdorff and $f \in \mathscr{K}(X;E)$ , one does not necessarily have $\int f d\mu \in E$ (Exer. 1; cf. No. 3).

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

Cited by

Proposition 1, § 3

Cites

Nothing.