Definition:Random Variable/Discrete

Definition
Let $$\mathcal E$$ be an experiment with a probability space $$\left({\Omega, \Sigma, \Pr}\right)$$.

A discrete random variable on $$\left({\Omega, \Sigma, \Pr}\right)$$ is a mapping $$X: \Omega \to \R$$ such that:
 * $$(1)$$ The image of $$X$$ is a countable subset of $$\R$$;
 * $$(2)$$ $$\forall x \in \R: \left\{{\omega \in \Omega: X \left({\omega}\right) = x}\right\} \in \Sigma$$

Alternatively (and meaning exactly the same thing), $$(2)$$ can be written as:
 * $$(2)$$ $$\forall x \in \R: X^{-1} \left({x}\right) \in \Sigma$$

where $$X^{-1} \left({x}\right)$$ denotes the preimage of $$x$$.

Note that if $$x$$ is not the image of any elementary event $$\omega$$, then $$X^{-1} \left({x}\right) = \varnothing$$ and of course by definition of event space as a sigma-algebra, $$\varnothing \in \Sigma$$.

Discussion
The meaning of condition $$(2)$$ can be explained as follows:

Suppose $$X$$ is a discrete random variable. Then it takes values in $$\R$$. But we don't know what the actual value of $$X$$ is going to be, since the outcome of $$\mathcal E$$ involves chance.

What we can do, though, is determine the probability that $$X$$ takes any particular value $$x$$.

To do this, we note that $$X$$ has the value $$x$$ iff the outcome of $$\mathcal E$$ lies in the subset of $$\Omega$$ which is mapped to $$x$$.

But for any such element $$x$$ of the image of $$X$$, the preimage of $$x$$ is an element of $$\Sigma$$.