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.