TITLE: Measure on a separable Hilbert space
QUESTION [6 upvotes]: Let $H$ be a real separable Hilbert space.
Is it true that there exist a probability space $(\Omega, \mu)$ and a measurable function $\pi\colon \Omega \to H$ such that for any $h \in H$ we have
$$
e^{-\lVert h\rVert^2}=\int_\Omega e^{i\langle h,\pi(\omega)\rangle}d\mu(\omega) \ \ ?
$$

REPLY [2 votes]: In Araujo and Giné book The central limit theorem in Banach spaces, there is an exercise which consists in showing that the map $\phi\colon x\mapsto \exp\left(-\frac{\lVert x\rVert^2}2\right)$ from $H$ to $\Bbb R$ is positive definite (in sense of the statement of Bochner theorem) but it's not the characteristic functional of a random variable. 
Indeed, considering $(\eta_j,j\geqslant 1)$ a sequence of i.i.d. Gaussian random variables and $X:=\sum_{j\geqslant 1}\eta_je_j$, where $(e_j,j\geqslant 1)$ is an orthonormal basis of $H$, we have, by uniqueness theorem in separable Hilbert spaces that $\phi$ would be the characteristic function of $X$. However, the sequence $\left(\sum_{j=1}^n\eta_je_j,n\geqslant 1\right)$ is not tight.