Increasing functions not necessarily continuous are measurable

Question

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be an increasing function (does not have to be continuous). Show that $f$ is measurable.

I'm having a problem proving this if the function is not continuous. Any help would be greatly appreciated.

Answer 1

$\{x: f(x) \leq a\}$ is an interval for any real number $a$. In fact, it is either an interval of the type $(-\infty,b)$ or an interval of the type $(-\infty,b]$. To see this all have to show is that if $f(x) \leq a$ then $f(y) \leq a$ for any $y \leq x$. Note: to show that $f$ is measurable it is enough to show that $\{x: f(x) \leq a\}$ is measurable for each $a$ because sets of the type $(-\infty,a]$ generate the Borel sigma algebra.