I am trying to figure out the following problem in measure theory and am stuck. It seems like it should be very easy, so I must be missing something.
Let $g: \mathbb{R} \to \mathbb{R}$ be a mapping of $\mathbb{R}$ onto $\mathbb{R}$ for which there is a constant $c > 0$ for which
$ |g(u) - g(v)| \geq c \cdot |u-v| \text{ for all } u, v \in \mathbb{R}. $
(Note to avoid confusion: this function is NOT Lipschitz and not supposed to be.)
Show that if $f: \mathbb{R} \to \mathbb{R}$ is Lebesgue measurable, then so is the composition $f \circ g$.
I see that we need to show that $g$ maps measurable sets to measurable sets. I know how to show $g$ is injective and that bounded sets are mapped to and from bounded sets... but I'm not sure where to go from there.
I'd appreciate a nudge in the right direction. Please do not give away the whole problem, if possible.