If $f,g: D \rightarrow \mathbb{R}$ are measurable, show $\{f>g\}=\{x \in D \mid f(x) > g(x)\}$ is measurable.
I'm new to this whole meausure theory thing and am gonna need some help...
$f,g$ are both measurable, so given any open set $U \subset \mathbb{R}$, both $f^{-1}(U)$ and $g^{-1}(U)$ are measurable. Therefore, it'd be nice if the image of $\{f>g\}=\{x \in D \mid f(x) > g(x)\}$ was open, because then I could just take the preimage and boom, it must be a measurable set, ha, but this must not be it. Can anyone lend some advice?