Sorry, I wrote a wrong assertion and I remove it.

Let my answer be as above, namely, if $\mu$ has no point masses, then we get squared moduli of the limits of the boundary values of the Cauchy tramsforms of $\mu$. For the purely point part of $\mu$ one has to add $\sum\left|\frac{\mu(\{z_n\})}{z-z_n}\right|^2$.

If $\mu>0$, the arithmetical means seem to tend to $2\int\frac{d\mu(\xi)}{|\xi-z|^2}$ (which may be infinite). I don't know where this could be proved, I will be grateful for references, if any.

Sorry, I wrote a wrong assertion and I remove it.