What is $\Omega$ in stochastic processes of the form: $X:[0,\infty )\times \Omega \to {\mathbb {R}}$?

Question

I understand that the $[0,\infty )$ interval is the time, but what are the elements of $\Omega$? Functions?

If yes, how does a sigma algebra generated by functions look like? And how does a probability measure on such a sigma algebra look like?

Answer 1

$\Omega$ is just your standard sample space. This could be coin tosses, for example. A non time-dependent random variable, $Y$ say, is a map $Y : \Omega \to \mathbb{R}$ (with the necessary measurabilit property). In this case, we have, for each $t \in [0,\infty)$, a random variable $X_t : \Omega \to \mathbb R$. Considering the process $X = (X_t)_{t\ge0}$, we have $X : [0,\infty) \times \Omega \to \mathbb R$.

Hopefully this helps. :)