Original question: $f$ is a Borel measurable function on $(0,1]$ such that $\int_{(\frac{k}{n},\frac{k+1}{n}]} f = 0$ for all $n\in \mathbb{N}$ and $0 \le k \le n-1 $ . Then to show that $f$ is zero almost everywhere with respect to Borel measure on $(0,1]$ .
I want to try proving that $\int_E f = 0$ for every Borel measurable $E \subset (0,1]$. For that I want to approximate Borel sets from above by finite disjoint unions of sets of the form $(\frac{k}{n}, \frac{k+1}{n}] $
Is this possible? If so can someone provide me with a hint? I know I can't approximate from below because set of irrationals can't be approximated from below by these kind of sets.