Remark

— 1) Let $E$ be a finite-dimensional vector space, and let $\mu$ be a positive measure on $E$ of mass 1, such that every linear form on $E$ belongs to $\mathscr{L}^2(E, \mu)$. One defines an element $m$ of $E$ and a positive quadratic form $V$ on $E'$ by the formulas

In the traditional terminology of Probability Theory, $m$ is called the mean and $V$ the variance of $\mu$; $\mu$ is said to be centered if $m = 0$.

Now let $a$ be an element of $E$ and $Q$ a positive quadratic form on $E'$. Let us denote by $\Gamma_{a,Q}$ the image of the measure $\Gamma_Q$ under the translation $x \mapsto x + a$. It is easily seen that $\Gamma_{a,Q}$ is a positive measure on $E$ of mass 1, with Fourier transform $x' \mapsto e^{i\langle a, x' \rangle - \frac{1}{2} Q(x')}$ and mean $a$. Moreover, Prop. 6 below implies that $Q$ is the variance of $\Gamma_{a,Q}$. One traditionally says that $\Gamma_{a,Q}$ is the Gaussian measure with mean $a$ and variance $Q$, and that $\Gamma_Q = \Gamma_{0,Q}$ is the centered Gaussian measure with variance $Q$. Since we shall only be considering centered Gaussian measures, we shall omit this qualifier.

2) Let $Q$ be a quadratic form on the dual $E'$ of a locally convex space $E$. If there exists a promeasure on $E$ with Fourier transform $e^{-Q/2}$, the quadratic form $Q$ is necessarily positive: for, the function $e^{-Q/2}$ is bounded on $E'$; therefore, for every $x' \in E'$, the function $t \mapsto e^{-t^2 Q(x')/2} = e^{-Q(tx')/2}$ on $\mathbf{R}$ is bounded, whence $Q(x') \geq 0$.

3) The dual of $\mathbf{R}$ is canonically isomorphic to $\mathbf{R}$ and the positive quadratic forms on $\mathbf{R}$ are the functions of the form $t \mapsto at^2$ with $a \geq 0$. Therefore, there exists for every $a \geq 0$ one and only bounded measure $\gamma_a$ on $\mathbf{R}$ whose Fourier transform is equal to the function $t \mapsto e^{-at^2/2}$; by an abuse of language, $\gamma_a$ is said to be the Gaussian measure on $\mathbf{R}$ with variance $a$.

The Fourier transform of $\gamma_0$ is the constant 1, whence $\gamma_0 = \varepsilon_0$ (unit mass at the origin of $\mathbf{R}$). Suppose $a > 0$ and denote by $u_a$ the linear mapping $x \mapsto a^{1/2} x$; then $\mathscr{F}\gamma_a = \mathscr{F}\gamma_1 \circ {}^t u_a$, whence $\gamma_a = u_a(\gamma_1)$. Lemma 2 shows that $\gamma_1$ is the measure with density $x \mapsto (2\pi)^{-1/2} e^{-x^2/2}$ with respect to Lebesgue measure; from this, one easily deduces

The image of a Gaussian promeasure under a continuous linear mapping is a Gaussian promeasure. More precisely, we have the following result: