Reference for Stieltjes transform of measure (Sokhotski-Pemelj)

Question

The Stieltjes transform of a probability measure $\mu$, supported on the real line, with $d\mu(x)=w(x)dx$ is defined as $$S_\mu(z)=\int_{\mathbb{R}}\frac{d\mu(t) }{t-z}=\int_{\mathbb{R}}\frac{w(t) }{t-z}dt.$$ I know that the following holds for any $x\in \mathbb{R}$: $$S_+(x)-S_{-}(x)=-2\pi iw(x), $$ with $$\lim\limits_{\epsilon\rightarrow 0+}S(x+i\epsilon)=S_+(x)\quad \text{ and }\quad \lim\limits_{\epsilon\rightarrow 0+}S(x-i\epsilon)=S_-(x). $$ This result is known to me as the Sokhotski-Pemelj lemma, but I can't find a good reference for this theorem. When I was looking online, the best I found was this pdf, but I don't know what book it comes from. Any help would be appreciated.

Answer 1

After looking a bit further, I found that the lemma is proven in this book: Boundary Value Problems by F. D. Gakhov in section I.4.2.