Random variables and Inverse maps

A random variable is a real valued function with domain $\Omega$ which has an extra property called measurability that allows us to make probability statements about the random variable.

Suppose $\Omega$ and $\Omega'$ are two sets.  Often $\Omega' = \mathbb{R}$.  Suppose
$$X:\Omega \mapsto \Omega',$$  Then $X$ determines an inverse map (a set valued function)
$$X^{-1}: \mathcal{P}(\Omega')\mapsto \mathcal{P}(\Omega)$$ defined by
$$X^{-1}(A') = \{\omega \in \Omega : X(\omega) \in A'\}$$ for $A' \subset \Omega'$.

$X^{-1}$ preserves complementation, union and intersections.