Prove that
If $f$ is a monotone function which defined on interval, then $f$ is measurable.
Proof: If $f$ is increasing and define on interval, then the set $A=\left\{x:f(x)>a\right\}$ will be an interval for all a, and it's measurable since each interval is measurable.
Does this an accepted proof? Need I to show that A is an interval? If yes; how can I prove that?
Suppose $f$ is increasing. $A$ is an interval; because $x\in A$ and $y >x$ implies $f(y) \geq f(x) >a$ so $y \in A$. This means $A$ contains all points from its infimum to $\infty$, so $A$ is an interval. A similar argument works if $f$ is decreasing.