Random Variable

A random variable $X$ is a measurable function from the sample space $\Omega$ to $\R$ $$ X : \Omega \to \R $$ that is, the inverse of any Borel Set in $\R$ is $\mathcal{F}$-measurable: $$ X^{-1} (A) = \{\omega : X(\omega) \in A \} \in \mathcal{F} \quad \forall A \in \R $$