# If the integral of a locally bounded function is zero for all domains it is zero

**Theorem :** Given a locally bounded function $f$, if its integral is zero,

for any domain $\Omega \subset \mathbb{R}^n$, then $f = 0$ almost everywhere.

**Proof :** For a bounded function, we have

As our integral is the same for every domain, we can always pick one in which our function is bounded around the appropriate point. We can also pick a smaller domain $\Omega'$ such that $2 \mu(\Omega') = \mu(\Omega)$, to get