When is the probability measure on the "direct product" via the Kolmogorov extension theorem 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.

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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.