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?



When is the probability measure on the "direct product" via the Kolmogorov extension theorem supported on the "direct sum"?
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](https://en.wikipedia.org/wiki/Kolmogorov%27s_three-series_theorem#Statement_of_the_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](https://en.wikipedia.org/wiki/Kolmogorov%27s_zero%E2%80%93one_law#Formulation). More specifically, this measure is $1$ if highlighted conditions 1 and 2 hold, and this measure is $0$ otherwise.


