Let $\displaystyle (\Omega , \mathcal{F} , \mathbb{P} ) $ be a probabilty space and $\displaystyle X: \ \Omega \rightarrow \ \bar{\mathbb{R}} $ be a random variable withdistribution $\displaystyle \mu $. Show that $\displaystyle \mu$ is aprobabilty measureon $\displaystyle \bar{\mathcal{B}} $. That is, verify the three axioms of probabilty for $\displaystyle \mu $, using the collection of "events" on $\displaystyle \bar{\mathcal{B}} $.