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,

\begin{equation} \int_\Omega f = 0 \end{equation}

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

Proof : For a bounded function, we have

\begin{equation} \mu(\Omega) \inf f\leq \int_\Omega f \leq \mu(\Omega) \sup f \end{equation}

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