Stieltjes transform of a measure ($g_{\mu }(z)={\int }_{\mathbb{R}}(x-z{)}^{-1}d\mu (x)$)

For a finite positive measure on the real line, this convention for its Stieltjes transform is analytic off the real line and has positive imaginary part in the upper half-plane if the measure is nonzero. Some sources use $(z-x{)}^{-1}$ instead, changing the sign. The transform of an empirical spectral measure equals the normalized trace of the Stieltjes matrix resolvent.