Gaussian Measure on a Hilbert Space (Segal)

Segal’s normal distribution over $M$ is a probability space $(N,\mathfrak R,n)$ such that each tame function $f(x)=\bar f(Px)$ has expectation given by the finite-dimensional Gaussian integral on $\mathrm{ran}(P)$.

Key properties (paper use):

• Defines $L_p(M,n)$ spaces used throughout §3.
• Quasi-invariance under $T$ holds exactly for $T\in \mathrm{rGL}(M)$ (see restricted general linear group ).

Example: In finite dimensions, this is the standard density $\propto e^{-\|x\|^2/2c}\,dx$.