integrability and distributions

Question

Let $G$ be a bounded open set in $\mathbb{R}^{n}$ ($n\geq1$), and $f$ a measurable function defined on $G$. Suppose for all test functions $\phi$ with support in $G$, we have \begin{equation} \int_{G}f(x)\Delta \phi(x)dx>-\infty,\end{equation} where $\Delta$ is the Laplace operator. Can we conclude that $$\int_{G}f(x)dx>-\infty?$$

Answer 1

No, you can't. $G=(0, 1)$ and $f(x) = - 1/x$ is a counterexample.