Existence and weak-type inequalities for Cauchy integrals of general measures on rectifiable curves and sets

If $\mu$ is a finite complex Borel measure and $\Gamma$ a Lipschitz graph in the complex plane, then for $\lambda > 0$

$$\left\vert {\left\{ {z \in \Gamma :\mathop {\sup }\limits_{\varep... ...\right\vert \leqslant c(\Gamma ){\lambda ^{ - 1}}\vert\vert\mu \vert{\vert _1}.$$

It follows that for any finite Borel measure $\mu$ and any rectifiable curve $\Gamma$ the finite principal value

$$\mathop {\lim }\limits_{\varepsilon \downarrow 0} \int_{\vert\zeta - z\vert \geqslant \varepsilon } {{{(\zeta - z)}^{ - 1}}d\mu \zeta }$$

exists for almost all (with respect to length) $z \in \Gamma$.