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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!

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up vote 3 down vote accepted

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.

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Perfect! That was what i figured, but learning new theory while guessing can sometimes make you quite dizzy. –  Henrik Apr 2 '12 at 19:16
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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 $X\in \mathscr F|\mathscr.B(\mathbb R)$

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Thank you! I do though know about those "tricks", just not the trick about writing that it belongs to the sigma-algebra (and isn't it by the way a horrible notation?). –  Henrik Apr 2 '12 at 19:20
    
@Henrik It is not, you just should get used to it. For example in "Markov Chains" by Revuz he considers them on a (fixed) measurable space $(E,\mathscr E)$ and explicitly writes that $A\in\mathscr E$ and $1_A\in \mathscr E$ both means that the set $A$ is measurable, although in the first statement we talk about the set and in the second - about the (indicator) function. –  Ilya Apr 2 '12 at 19:22
    
Wow, okay! I'm puzzled why I haven't seen it before then, but I see from your example that it does look kind of neat :) thanks again! –  Henrik Apr 2 '12 at 19:28
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