Let $f(z) = \int_\mathbb{C} \log |w - z| \,d\mu(w)$ for some measure $\mu$. How does one show that $\mu = \Delta f / 2\pi$ in the sense of distributions?
As a related question, how can one show that the distributional Laplacian $\Delta f$ of $$f(z) = \int_0^{2\pi} \log|e^{i\theta} - z|\,d\theta$$ is simply Lebesgue measure restricted to the unit circle?