Let $E$ be measurable and define $f:E\rightarrow\mathbb{R}$ such that $\{x\in E : f(x)>c\}$ is measurable for all $c\in\mathbb{Q}$, is $f$ measurable?
There are a number of equivalent definitions for the measurability of a function and the most obvious one would be to show that $\{x\in E : f(x)>c\}$ is measurable for all $c\in\mathbb{R}$.
Thus my strategy has been to consider an arbitrary irrational $y$ and use the density of the rationals in the reals to show that there exists some open set $O$ such that $m^*(O-\{x\in E : f(x)>y\})< \varepsilon$. I would do this by choosing some rational $q
Is there a better definition of the measurability of a set I should use? or does the statement actually not imply measurability? Thanks.