Discrete random variables

A discrete random variable can be defined even on a continuous sample space. For instance, on $S=[0,1]$, the constant function $X(\omega)=10$ for all $\omega\in S$ is (trivially) a discrete random variable — its range is the single point $\{10\}$. Less trivially, on $S=[0,20]$ the two-valued step function

$X(\omega)=\begin{cases}1 & \omega\in[0,10)\\ 2 & \omega\in[10,20]\end{cases}$ …