Stochastic variable belonging to sigma-field I'm studying Markov Processes in Rick Durrett - Probability: Theory and Examples and he's doing something I simply don't understand, though I reckon it's probably quite simple. Here goes (an example from introducing conditional expectations):
Given a probability space $\left(\Omega,\mathcal{F}_{0},P\right)$ a $\sigma\text{-field}\,\mathcal{F}\subset\mathcal{F}_{0}$ and a random variable $X\in\mathcal{F}_{0}$...
What does it mean for $X\in\mathcal{F}_{0}$? I mean, the image of X has to be Borel, right?
It belonging to a $\sigma$-algebra in our probability space doesn't make sense to me.
Hope someone will help,
Henrik
p.s. Wow the math on this site works good!
 A: Durrett's notation (I think he mentions this quietly in the very first chapter) is that $X\in \mathcal{F}_0$ means $X$ is $\mathcal{F}_0$ measurable. So in general if you see a random variable being an element of a sigma algebra, then he just means it's measurable w.r.t. that sigma-algebra.  
A: Although Sam has perfectly answered your question, there may be something to add. FWIW, I hope you are aware of notion of measurability - given two measurable spaces $(\Omega,\mathscr F)$ and $(E,\mathscr E)$ the function $X:\Omega\to E$ is called measurable if $X^{-1}(\mathscr E)\subset\mathscr F$. Usually it is denoted as
$$
X:(\Omega,\mathscr F)\to(E,\mathscr E)
$$
which is quite hard to write always - so more simple notation is $X\in \mathscr F|\mathscr E$ where the domain $\Omega$ and the codomain $E$ are omitted - so it refers to the case when they don't vary or assumed to be understood from the context (since e.g. the $\sigma$-algebra $\mathscr F$ itself carries an information that it is defined exactly on $\Omega$). Furthermore, for real-valued random variables it is a very usual case that $\mathscr E$ is a Borel $\sigma$-algebra on $\mathbb R$, so there is no point to write it every time - that's why  we write $X\in \mathscr F$ instead of writing $X\in \mathscr F|\mathscr.B(\mathbb R)$
