I´m a beginner in more advanced probability and measure theory and there is this definition that I simply can´t understand. It says, a random variable is a function $X\colon\Sigma\to \mathbb R$ with the property that the set $\{\sigma \subseteq \Sigma": X(\sigma)\in B\}$ belongs to $\mathcal F$ for each Borel set $B$. ($\mathcal F$ is a $\sigma$-algebra).
Does it mean that $X(\sigma)$ should be contained in every possible Borel set for all values of sigma? Or is it the other way around, that we start by "looking" at each Borel set to find out which values of sigma that makes $X(\sigma)$ belong to each one of the Borel sets, and then finally "look" if all these sigmas belong to the sigma algebra $\mathcal F$?
I've had a really hard time trying to find out what this definition really means so your answers would be much appreciated. Thanks in advance! (Sorry, but I´m not used to Latex).
Best Answer
The definition says that $X$ is $F$-measureable, which (as Alex pointed out) amounts to requiring the sublevel sets $\{\omega:X(\omega)\le \alpha\}$ to be in $F$ for every real number $\alpha$. (Equivalently, one can ask for this property with $<\alpha$, $\ge\alpha$, or $>\alpha$).
To see why we might care about such a thing, consider the special case of $X$ taking values $0$ and $1$ only. Then the expected value of $X$ is the measure of the set $\{\omega:X(\omega)=1\}$ which is not defined unless this set is in $F$. In general, measurability is needed to compute the expected value, variance, and other quantities describing $X$.
Studying the subject further, one finds that measurability can also express the idea of a random variable being independent of something.