Problem: Let $E$ be a Borel set, and $f: E \rightarrow \mathbb{R}$ a Lebesgue measurable function, and $\int_{B \cap E} f dm = 0$ for every open ball centered at points in $E$. Show that $f = 0$ $\mu$-a.e. on $E$.
Attempt: Denote $F^+ = \{x: f(x) > 0\}$ and $F^- = \{x: f(x) < 0\}$; we focus on $F^+$ WLOG. We can define $F^+_n = \{x: f(x) > 1/n\}$. Then $F^+_n$ is Lebesgue measurable. Moreover, $\frac{1}{n}m(F^+_n) \le \int_{F^+_n \cap E} f$. Now I have trouble finding an open ball intersecting $E$ that bounds $\int_{F^+_n \cap E} f$. I tried forming $\cup_i B_i$ such that $m(F^+_n) \lt m(\cup_i B_i) + m(N)$ where $m(N) = \epsilon$ for some $\epsilon > 0$, but I cannot directly say $\int_{F^+_n \cap E} f \le \int_{\cup_i B_i \cap E} f$ since on the set $N$, $\int_{N \cap E} f$ could well be a negative number. Now how should I proceed? Is there an easier way to solve this problem? Any help is appreciated!