Gaussian Measure on a Hilbert Space (Segal)

Let $M$ be a real Hilbert space. A normal distribution over $M$ in Segal's sense is a probability space $(N,\mathfrak R,n)$ carrying a real-linear map

W:M\longrightarrow L^2(N,n)

such that the random variables $W(x)$ are jointly centered Gaussian and

\int_N W(x)W(y)\,dn=\langle x,y\rangle_M \qquad(x,y\in M).

Thus $W$ is an isometric embedding of $M$ into the real Gaussian subspace of $L^2(N,n)$ .

Finite-dimensional distributions

If $e_1,\ldots,e_m$ are orthonormal and

F=\bar F\bigl(W(e_1),\ldots,W(e_m)\bigr)

is integrable, then

\int_N F\,dn =(2\pi)^{-m/2}\int_{\mathbb R^m} \bar F(t_1,\ldots,t_m)e^{-\lVert t\rVert^2/2}\,dt.

This compatibility of all finite-dimensional Gaussian marginals is the integration rule for tame functions. A covariance $cI$ convention is obtained by replacing the right-hand covariance above by $c\langle x,y\rangle_M$ .

Why the sample space is not

M

If $M$ is infinite-dimensional, there is no countably additive Gaussian Borel probability measure on $M$ having covariance $I$ : the covariance operator of a Gaussian Borel measure on a Hilbert space must be trace class, whereas $I$ is not. The notation $L^p(M,n)$ used in this context therefore refers to the Gaussian probability-space realization, not literally to a standard Gaussian measure supported on $M$ .

Shale's setting

These Gaussian $L^p$ -spaces are used in §3 of Shale's paper. The relevant Gaussian measure class is quasi-invariant under the restricted general linear group.