Principal values for the Cauchy integral and rectifiability

We give a geometric characterization of those positive finite measures $\mu$on ${\mathbb C}$ with the upper density $\limsup_{r \to 0} \frac{\mu(\{\xi:\vert\xi-z\vert\leq r\})}{r}$ finite at $\mu$-almost every $z\in{\mathbb C}$, such that the principal value of the Cauchy integral of $\mu$,

exists for $\mu$-almost all $z\in{\mathbb C}$. This characterization is given in terms of the curvature of the measure $\mu$. In particular, we get that for $E\subset{\mathbb C}$, ${\mathcal H}^1$-measurable (where ${\mathcal H}^1$ is the Hausdorff $1$-dimensional measure) with $0<{\mathcal H}^1(E)<\infty$, if the principal value of the Cauchy integral of ${\mathcal H}^1_{\mid E}$ exists ${\mathcal H}^1$-almost everywhere in $E$, then $E$ is rectifiable.