Proposition 11

Tag 09T1, permanent. Cite it as [Bourbaki, Tag 09T1].

Integration, chapter IX, § 6. Promeasures and measures on a locally convex space

— Let E be a finite-dimensional vector space, $\mu$ a bounded (positive) measure on E, and $\Phi$ the Fourier transform of $\mu$ . The function $\Phi$ is continuous and of positive type on $E'$ .

The continuity of $\Phi$ follows from Prop. 9 of No. 8.

Let us show that $\Phi$ is of positive type. Let $x'_1, \ldots, x'_p$ be in $E'$ and $c_1, \ldots, c_p$ complex numbers. Then

\sum_{j,k} c_j \overline{c_k} \Phi(x'_j - x'_k) = \int_E \sum_{j,k} c_j \overline{c_k} e^{i \langle x, x'_j - x'_k \rangle} \, d\mu(x)

= \int_E \left| \sum_{j=1}^p c_j e^{i \langle x, x'_j \rangle} \right|^2 \, d\mu(x) \geq 0.

Q.E.D.

One can prove a converse known as Bochner's theorem: every continuous function on $E'$ of positive type is the Fourier transform of a bounded (positive) measure. (*) We shall assume this result for the rest of No. 12.

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

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Cites

Prop. 9