Proposition 6

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

Integration, IX, p. 81, § 6. Promeasures and measures on a locally convex space

Let E be a locally convex space, and $\mu$ a Gaussian measure on E, with variance Q. For every integer $n \geq 0$ and every $x' \in E'$ , one has the relations

\begin{align*} (16) & \quad \int_E |\langle x, x' \rangle|^n d\mu(x) = \pi^{-1/2} 2^{n/2} \Gamma\left( \frac{n+1}{2} \right) Q(x')^{n/2} \\ (17) & \quad \int_E \langle x, x' \rangle^{2n} d\mu(x) = \frac{(2n)!}{2^n n!} Q(x')^n \\ (18) & \quad \int_E \langle x, x' \rangle^{2n+1} d\mu(x) = 0 . \end{align*}

In particular,

(19) \quad \int_E \langle x, x' \rangle^2 d\mu(x) = Q(x') \qquad (x' \in E') .

If these formulas are true for an element $x'$ of $E'$ , then they are true for all of its multiples $t \cdot x'$ (with $t$ real). We may therefore content ourselves with establishing them when $Q(x')$ is equal to 0 or 1.

a) Suppose $Q(x') = 0$ . The measure $x'(\mu)$ is equal to $\gamma_0 = \varepsilon_0$ , therefore $x'$ is zero $\mu$ -almost everywhere; the formulas (16) to (19) are then obvious.

b) Suppose $Q(x') = 1$ , whence $x'(\mu) = \gamma_1$ . Then

\int_E |\langle x, x' \rangle|^n d\mu(x) = \int_{\mathbf{R}} |t|^n d\gamma_1(t) = (2\pi)^{-1/2} \int_{\mathbf{R}} |t|^n e^{-t^2/2} dt

and (16) follows immediately from (6) (No 4, Lemma 1). Similarly, formulas (17) and (18) follow from (7) and (8). Finally, (19) is obtained by setting

n = 1

in (17).

We can now prove a converse of the Cor. of Prop. 5.

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

Proposition 6

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