Uniform convergence of a complex integral

Question

Let E a compact in the complex plane and U his complement that is connected. Let μ a complex measure on the boundary of U. I don't undestand why the integral of |1/(z-t)|dμ(t) in boundary of U converges in a.e in the plane. (hint:Fubini's theorem)

Thanks

Answer 1

Hint: For $r>0,$ is

$$\int_{\{|z| < r\}}\int_{\partial U}\frac{1}{|\zeta - z|}\,d|\mu|(\zeta)\, dm_2(z) < \infty?$$

Here $m_2$ is Lebesgue measure on $\mathbb C.$