Proposition 6

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

Integration, III, p. 37, § 3. Integrals of continuous vector-valued functions

For every continuous mapping with compact support $f$ of $X$ into a Hausdorff locally convex space $E$ , every continuous semi-norm $q$ on $E$ and every measure $\mu$ on $X$ such that $\int f d\mu \in E$ ,

q \left( \int f d\mu \right) \leq \int (q \circ f) \ d|\mu|.

Let $D$ be the set of $z \in E$ such that $q(z) \leq 1$ ; $D$ is closed, convex and contains 0, therefore $D = D^{o o}$ (TVS, II, § 6, No. 3, Cor. 3 of Th. 1). It therefore suffices to prove that for every $z' \in D^o$ ,

\left| \left\langle \int f d\mu, z' \right\rangle \right| \leq \int (q \circ f) \ d|\mu|;

since

\left\langle \int f d\mu, z' \right\rangle = \int \langle f, z' \rangle d\mu,

and since, by the definition of $D^o$ ,

|\langle f(x), z' \rangle| \leq q(f(x))

for all $x \in X$ , the desired inequality follows from the inequality (13) of § 1, No. 6.

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

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Cites

§ 1, No. 6