In my script there is a proof for „ Every monotonic function $f:\mathbb{R} \rightarrow \mathbb{R}$ is measurable“, I don‘t understand.
Let $a\in \mathbb{R}$. We want to show $A=\{ x\in \mathbb{R} |f\left(x\right) >a\}$ is measurable (which is equivalent to $f$ is measurable).
Let f be increasing. Is $f\left( x\right) >a$ and $y\geq x$, then $f\left( y\right) \geq f\left( x\right) >a$. Thus $A$ is an interval either of the form $[ c,\infty [$ or $] c, \infty[$, thus measurable.
What i don‘t understand is, why does $A$ have to be an interval?
Isn‘t $A$ only an interval, if $f$ is continuous?
On the same page there is another statement, without proof.
A monotonic function $f:\mathbb{R} \rightarrow \mathbb{R}$ is integrable, iff $f\left( x\right) =0$ for almost every $x\in \mathbb{R}$.
Is there a simple proof for that?
Notice The intervals $[c,\infty[$ or $]c,\infty[$ are on domain, NOT in range. And $y$ can take any real greater than $x$. So they produce an interval in domain, without need to continuity ,as claimed.