Let $(\Omega, \mathcal{F}, P)$ be a probability space. 

Sometimes, the convention is used that a random variable is a map from a probabilty space to $\mathbb{R}$, but let's not adopt this. So let $X$ be a random variable with range $E$, 

so $X \colon \Omega \rightarrow E$. 

The question now is: Why must $E$ be measurable? What does it mean for a range not to be measurable?