Kolmogorov Extension Theorem – When is the Probability Measure on the Direct Product Supported on the Direct Sum

Question

Let me restrict to the case of Hilbert spaces, which seem simplest.

Let $\{H_n\}$ be a sequence of (possibly infinite dimensional) Hilbert spaces and $\{ \mu_n \}$ be a sequence of Borel probability measures on them. Then, by the Kolmogorov extension theorem, it is straightforward to see that there exists a Borel probability measure $\mu_\infty$ on the "direct product space"
\begin{equation}
\prod_{n=1}^\infty H_n
\end{equation}
such that the "projection" of $\mu_\infty$ onto any finite direct product of $H_n$'s is equal to the tensor product of $\mu_n$'s for the corresponding indices.

Note that the space $\prod_{n=1}^\infty H_n$ is equipped with the product topology, but it is not a Hilbert space itself. An usual construction of the Hilbert space is the "orthogonal direct sum", which is a subspace of $\prod_{n=1}^\infty H_n$ and defined by
\begin{equation}
\bigoplus_{n=1}^\infty H_n = \Bigl\{ (x_n) \in \prod_{n=1}^\infty H_n \mid \sum_{n=1}^\infty \lVert x_n \rVert^2 < \infty\Bigr\}.
\end{equation}

Now, my quesiton is that, under which conditions is $\mu_\infty$ supported on $\bigoplus_{n=1}^\infty H_n$? Or at least is it always the case that $\bigoplus_{n=1}^\infty H_n$ is of nonzero measure with respect to $\mu_\infty$?

The Minlos theorem keeps coming up in my head, but the situation now deals with possibly infinite dimensional Hilbert spaces. So, I am quite stuck.

Could anyone help me?

Answer 1

It is convenient to restate the main question as follows:

For each natural $n$, let $X_n$ be a random vector in $H_n$. Suppose that the $X_n$'s are independent. Under what further conditions do we have $$\sum_{n=1}^\infty\|X_n\|^2<\infty\tag{1}\label{1}$$ almost surely (a.s.)?

The answer to this question follows immediately from Kolmogorov's three-series theorem and is as follows:

We have \eqref{1} if and only if for some real $A>0$ (or, equivalently, for each real $A>0$) the following two conditions hold:

1. $\sum_{n=1}^\infty P(\|X_n\|\ge A)<\infty$ and
2. $\sum_{n=1}^\infty EY_n <\infty$, where $Y_n:=\|X_n\|^2\,1(\|X_n\|\le A)$.

(Note that $\sum_{n=1}^\infty Var\,Y_n\le\sum_{n=1}^\infty EY_n^2 \le A^2\sum_{n=1}^\infty EY_n <\infty$ given condition 2 above.)

For instance, if $P(\|X_n\|=2^n)=1/n^2=1-P(\|X_n\|=0)$, then \eqref{1} holds.

---

As for your additional question: "is it always the case that $\bigoplus_{n=1}^\infty H_n$ is of nonzero measure with respect to $\mu_\infty$?" -- this measure is always either $0$ or $1$, by Kolmogorov's zero–one law. More specifically, this measure is $1$ if highlighted conditions 1 and 2 hold, and this measure is $0$ otherwise.