If $E\subset \mathbb{R}$ is a set Lebesgue Measurable and $f:E \rightarrow \mathbb{R}$ a monotone function, show that $f$ is measurable.
I'm trying for hours with no progress.
If $E\subset \mathbb{R}$ is a set Lebesgue Measurable and $f:E \rightarrow \mathbb{R}$ a monotone function, show that $f$ is measurable.
I'm trying for hours with no progress.
Every monotone function is even Borel measurable, and in particular Lebesgue measurable. To see this, first let $f$ be an increasing function on $\mathbb{R}$. Let $a \in \mathbb{R}$.
Define $F_a=\{ x\ \in \mathbb{R} : f(x) \leq a \}$, and we want to show $F_a$ is a measurable set, from which we will conclude f is measurable.
Now if $F_a=\varnothing $ then obviously $F_a$ is measurable, and we are done.
So assume $F_a\not =\varnothing $, and let $x_0=\sup F_a$, There are 3 different options:
1)if $x_0=\infty$, then $F_a=\mathbb{R}$ is measurable.
2)if $x_0 \in F_a$ then $F_a=(-\infty,x_0]$.
3) if $x_0 \not \in F_a$ then $F_a=(-\infty,x_0)$.
and in any case $F_a$ is a borel set, and so f is measurable.
Now for f decreasing you may note that $-f$ is an increasing function, so it is measurable, and therefore $f$ is measurable too.