Random variables definition I am looking at my lecture notes and would like to ask for a clarification of the following property. So I have a definition of a random variable:
" $X(w) \in \mathbb{R}$, i.e. $X:\Omega \rightarrow \mathbb{R}$ and $\omega \in \Omega$ with the property that $\forall x \in \mathbb{R}, \left\{\omega \mid X(\omega)\leq x  \right\} \in \Sigma$"
where $\Sigma$ is the sigma-field.
I don't quite follow that last part. What does it mean?

The last part implies that $X$ is a measurable function from $(Ω,Σ)$ to $(\mathbb R,\mathcal B(\mathbb R))$

$$ X^{-1}((-\infty,x)):=\{ω\in Ω\mid X(ω)\le x\}\in \Sigma ,\;\;\forall x\in \mathbb R $$

Now, 


*

*Why only sets of the form $(-\infty, x)$? Because they generate the Borel σ-algebra on $\mathbb R$ (so is suffices to look at them, i.e. to check measurability on the generating set.). And 

*Why has $X$ to be measurable? Because we want to calculate (or measure) probabilities of events (which are subsets of $Σ$.)

