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In Durrett's Probability:Theory and Examples page 205 section 4.1, it has the following notation $X\in \mathcal{F}_o$ (see the picture below). I'm confused about this notation as $X$ is a random variable, while $\mathcal{F}_o$ is a sigma-algebra. $\mathcal{F}_o$ is the collection of subsets of $\Omega$, so in order to write something like $a\in\mathcal{F}_o$, $a$ must be a subset of $\Omega$. How could $X$ be an element of it? (I didn't read the previous sections of this book, if you know there are explanations for this notation somewhere before that would be great too.) enter image description here

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2 Answers 2

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This is not exclusive to Durrett or that he invented this. When you need to clarify the sigma-algebra based on which the function is measurable, it is used in that way.

It all boils down to one's taste: some authors adopt this while others don't.

Erhan Çinlar, for example, uses this for brevity, as mentioned in $\rm[I]$ (emphasis mine) (in fact, he introduced this notation usage back in the first chapter):

Let $(\Omega, \mathcal H, \mathbb P) $ be a probability space. Let $\mathcal F$ be a sub–$\sigma$–algebra of $\mathcal H, $ and $X$ an $\bar{\mathbb R}$–valued random variable. As usual, we regard $\mathcal F$ both as a collection of events and as the collection of all $\mathcal F$–measurable random variables.

Coming to Durrett, he also clearly stated this usage $\rm [II, 1.2]:$

When we need to emphasize the $\sigma$–field, we will say that $X$ is $\mathcal F$–measurable or $X\in\mathcal F. $

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References:

$\rm [I]$ Probability and Stochastics, Erhan Çinlar, Springer Science$+$Business, $2011, $ sec. $\rm IV. 1.$

$\rm[II]$ Probability: Theory and Examples, Rick Durrett, $2010, $ Cambridge University Press.

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  • $\begingroup$ Thanks a lot! This is very helpful! $\endgroup$ Commented Apr 23 at 3:32
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He's pretty clearly using the notation $X\in{\cal F}$ for a random variable $X$ and $\sigma$-field ${\cal F}$ to say that $X$ is ${\cal F}$-measurable: for any Borel set $A$, the set $\{\omega\mid X(\omega)\in A\}$ is an element of ${\cal F}$. I can't find where in the previous chapters he says this, though.

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  • $\begingroup$ Thanks! This is very helpful! $\endgroup$ Commented Apr 23 at 3:32

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