Uniqueness of measures related to the Stieltjes transforms

As I mentioned in an earlier comment, The imaginary part of the Stieltjes transform $I_mu$ (in your notation) is $pi P*mu$, where $P$ is the Poisson kernel.

A short but concise presentation if the Stieltjes integral in Barry Simon's Comprehensive Course in Analysis, Volume 3 (Harmonic Analysis) section 2.5. Amongst all the different cool things one can find there is the following Theorem:

Theorem: Let $mu$ be a finite measure on $mathbb{R}$ and $F_mu(z)=intfrac{mu(dx)}{x-z}$ be its Stieltjes transform. Suppose $$ mu(dx) = fcdotlambda(dx) + mu_s(dx)$$ be its Lebesgue decomposition (absolute and singular decomposition with respect the LEbeshue measure $lambda$. Then

a. $frac{1}{pi}int operatorname{Im},F_mu(x+ ivarepsilon)g(x),dx xrightarrow{varepsilonrightarrow0+}int g(x)mu(dx)$ for any $gin mathcal{C}_{00}(mathbb{R})$ (functions of compact support).

b. For $lambda$-a.a. $xin mathbb{R}$, $$frac{1}{pi}lim_{varepsilonrightarrow0+}operatorname{Im} F_mu(x+ivarepsilon)= f(x)$$

c. There exists a measurable set $Esubsetmathbb{R}$ such that $lambda(E)=0=mu_s(mathbb{R}setminus E)$ such that $$ lim_{varepsilonrightarrow0+}operatorname{Im} F_mu(x+ivarepsilon)= infty,qquad xin E $$

d. For all $x_0inmathbb{R}$,

$$ lim_{varepsilonrightarrow0+}varepsilonoperatorname{Im} F_mu(x_0+ivarepsilon)= mu({x_0}) $$

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I hope this helps answer your question, or at least provides you with good reference that discusses interesting results and historical notes about the Stieltjes transform.

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