Definition:Random Variable/Discrete/Definition 1

Definition
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\struct {\Omega', \Sigma'}$ be a measurable space.

A discrete random variable on $\struct {\Omega, \Sigma, \Pr}$ taking values in $\struct {\Omega', \Sigma'}$ is a mapping $X: \Omega \to \Omega'$ such that:
 * $(1): \quad$ The image of $X$ is a countable subset of $\Omega'$
 * $(2): \quad$ $\forall x \in \R: \set {\omega \in \Omega: \map X \omega = x} \in \Sigma$

Alternatively, the second condition can be written as:
 * $(2): \quad$ $\forall x \in \R: X^{-1} \sqbrk {\set x} \in \Sigma$

where $X^{-1} \sqbrk {\set x}$ denotes the preimage of $\set x$.