I came across this theorem in one of my courses on Sobolev spaces. I am struggling to find a clear proof for it and I have not succeeded yet.
Some notation first:
$L_{loc}^{1}\left( \Omega \right) $ is the set of functions $f:\Omega\rightarrow\mathbb{R}$ that are locally integrable, that is \begin{equation*} L_{loc}^{1}\left( \Omega \right) =\left\{ f:\Omega \rightarrow \mathbb{R}% \mathbf{;}\exists \int_{K}\left\vert f\left( x\right) \right\vert dx<\infty ,\forall K\subset \Omega ,K~\text{compact set}\right\} \end{equation*}
$\mathcal{D}\left( \Omega \right)$ is the set of continuously differentiable functions, that is \begin{equation*} C_{0}^{\infty }\left( \Omega \right) =\left\{ \varphi :\Omega \rightarrow \mathbb{R}\mathbf{;~}\varphi \in C^{\infty }\left( \Omega \right) ~\text{ and }supp(\varphi )\subset \Omega ,\ supp(\varphi )\ \text{compact}% \right\} . \end{equation*}
This is the Theorem:
Let $\Omega \subset \mathbb{R}^{N}$ be an open set and $f\in L_{loc}^{1}\left( \Omega \right) $ so that \begin{equation*} \int_{\Omega }f\left( x\right) \varphi \left( x\right) dx=0,\quad \forall \varphi \in \mathcal{D}\left( \Omega \right) \end{equation*} Then, $f\left( x\right) =0$ a.p.t. $x\in \Omega $.