How to show that ${\omega \in \Omega : X(\omega) \neq Y(\omega)}$ is a measurable set, $X,Y$ being $\overline{\mathbb{R}}$-valued. How to show that $\{\omega \in \Omega : X(\omega) \neq Y(\omega)\}$ is a measurable set, $X,Y$ being $\overline{\mathbb{R}}$-valued. I mean, how to fix the case when $X$ and $Y$ = $\infty$? Here i can't pass to the $\{\omega \in \Omega : X(\omega) - Y(\omega) \neq 0\}$ because $\infty - \infty$ isn't defined. Thank you!

$\{\omega: X(\omega)=\infty, Y(\omega)<\infty\} =\bigcup_n  \{\omega: X(\omega)=\infty\}\bigcap \{\omega: Y(\omega)<n\}$ which is measurable. Simialrly, $\{\omega: Y(\omega)=\infty, X(\omega)<\infty\}$ is measurable. Can you finish?
