I need to get familiar with probabilities over separable Hilbert spaces. It's a bit difficult for me to think of something as a probability distribution over a space of function: I can't imagine how it could be defined, but in terms of finite-dimensional objects.
For example, let's consider the only probability distribution over a function space that I know of: the Gaussian Process. We say that
$f\sim GP(\mu(x),c(x,x'))$
if, for any finite set $X_n=\{x_1, \dots, x_n\}$, $\mathbf{f}=(f(x_1),\dots,f(x_n))$ is a random vector having the multivariate normal distribution $\mathcal{N}(\boldsymbol{\mu},\boldsymbol{\Sigma})$ with $\boldsymbol{\mu}=(\mu(x_1),\dots,\mu(_n))$ and $\boldsymbol{\Sigma}$ is a symmetric positive definite matrix with components $\Sigma_{ij}=c(x_i,x_j)$. Are there other practical examples of probability laws over function spaces? Are they always defined in terms of probability distributions of finite-dimensional vectors/matrices, or is it possible to define them directly in terms of infinite-dimensional objects?
Also, does it makes sense to talk about a CDF for infinite-dimensional objects? It seems to me that we can at least talk of a Cumulative Distribution Functional. For example, it could make sense to compute
$P(F(x)<g(x))$
where $F(x)$ is my random function and $g(x)$ is any element of the separable Hilbert space, by interpreting the above inequality in the natural sense. I.e., we ask ourselves which is the probability that $F$ is less than $g$ for all $x\in X$ where $X$ is the domain of the functions in our separable Hilbert space. Then the role of the CDF in finite-dimensional probability spaces would be taken by the functional $\mathcal{F}(g)=P(F(x)<g(x))$. I don't know how useful this construct would be, though: I don't even know if it makes sense to talk about continuous functionals (I seem to remember from Functional Analysis that you only considered continuous linear functionals, but this functional is obviously not linear).
As in my comment above, we can consider a sequence of i.i.d. random variables $\{X_i\}_{i=1}^{\infty}$ uniform over $[0,1]$, and define $g(t) = \sum_{i=1}^{\infty} \frac{1}{2^i}X_i \sin(2\pi i t)$, which is perhaps an example of what Pepe was talking about in his answer.
I have never seen an infinite dimensional PDF or CDF, I don’t think there is a need for that. The $\{X_i\}$ variables are mutually independent if $$ P[\cap_{i=1}^m \{X_i\leq x_i\}] = \prod_{i=1}^m P[X_i \leq x_i] $$ for all integers $m>0$ and all $(x_1, …, x_m) \in \mathbb{R}^m$. Then outcome-sets expressed by $g(t)$ are measurable if they are in the sigma-algebra generated by the $X_i$ variables.
So for example, what is $P[g(1)\leq 4.6]$? It can be shown that $\{g(1)\leq 4.6\}$ holds if and only if for all integers $j>0$ there is an integer $k_j>0$ such that $$\sum_{i=1}^m \frac{1}{2^i}X_i \sin(2 \pi i) \leq 4.6 + \frac{1}{j} \quad, \forall m \geq k_j $$
So, $$ \{g(1) \leq 4.6 \} = \cap_{j=1}^{\infty} \cup_{k=1}^{\infty} \cap_{m=k}^{\infty} \left\{ \sum_{i=1}^m\frac{1}{2^i}X_i\sin(2\pi i) \leq 4.6 + \frac{1}{j}\right\}$$ and so the probability of the complicated event $\{g(1)\leq 4.6\}$ is expressed by intersections and unions of basic events involving a finite number of the $X_i$ variables.