I can't quite see why the following integral should be finite. Given a compact set $F$, the complex plane $\mathbb{C}$ and a compactly supported and integrable function $\phi$ show that
$\int_{F}\int_{\mathbb{C}}\frac{|\phi(\tau)|}{|z-\tau|}dzd\tau<\infty$.
It looks to me that for each fixed $\tau$ for which $\phi(\tau)\neq0$ the contribution of the integral term $\int_{\mathbb{C}}\frac{|\phi(\tau)|}{|z-\tau|}dz$ is infinite and so if the support of $\phi$ has positive measure the integral should then be infinite. Why is it not so?