Suppose I have a random signal, like electrical noise, $X(t)$ such that the value of the signal at some instant $t_0$ is a random variable with known $\mathrm{p.d.f.}$ $f_{X(t_0)}(x)$ (I take for simplicity that the probability does not depend on time nor on previous values of the signal). With this in mind, there must exist some set $S$ of posible signals that can be generted by the random variables at each moment on time.
So my question (asuming that the previous statements are logic) is: exist a way to define/obtain a probability density distribution over that space? Such that integrating over a subset gives me the probability that I will observe a signal from that subset?
I think if exist such thing it will a functional $F[x]$ so that the probability should be calculated from a functional integral (probably difficult to compute, if not at all of course): $$P( X \in A) = \int_A \mathcal{D}x \ F[x] \qquad \forall A \subseteq S$$
I tried to figure out the form of $F[x]$ by multiplying $f_{X(t)}(x)$ for many diferents moments in discrete time (since are independent of each other) and then taking the limit into continumm, but I'm not sure about that, and maybe I'll post it in a separate question.
Is there a brach/field of mathematicts that deals with this type of problems?
Thanks in advance! And sorry for the bad english and the potential bad notation.