I am familiar with basic probability and measure theory, and would like to know if there is an easily accessible reference, where I can understand under what conditions something like the following makes sense: $$ w = (w_1, \dots, w_n, \dots) \sim N_\infty(0, \Sigma), $$ is an infinite dimensional sequence of Gaussians, and $\Sigma$ is a positive, self-adjoint operator on $\ell^2(\mathbb{N})$. I would like to understand things like when $\mathbb{E}[\langle w, A w\rangle_{\ell^2(\mathbb{N})}]$ is meaningful, exists and is finite, when $A$ is a positive operator on $\ell^2(\mathbb{N})$.
Is there a relatively accessible resource for this type of thing?