TITLE: Is a function determined by its integrals over open sets?
QUESTION [8 upvotes]: If $f \in L^1(\mathbb R)$ satisfies
$$
  \int_U f = 0
$$
for every open set $U \subset \mathbb R$, then is it true that $f = 0$ a.e.?

REPLY [7 votes]: Your assumption implies that $\int_E f = 0 $ for every compact set $E$. But since every positive measure set has a positive measure compact subset, none of the sets $\{x : f(x) > 0\}$, $\{x : f(x) < 0\}$ can have positive measure.