Harmonic measures supported on curves (Q1123286)

The following nice result is proved: Let \(\Omega_ 1\), \(\Omega_ 2\) be the two complementary domains of a Jordan curve \(\Gamma\) in the complex plane and let \(\omega_ 1\), \(\omega_ 2\) be the harmonic measures on \(\Gamma\) for the two regions \(\Omega_ 1\), \(\Omega_ 2\) with respect to two given points \(z_ 1\in \Omega_ 1\) and \(z_ 2\in \Omega_ 2\). Then (1) and (2) below are equivalent: (1) \(\omega_ 1\perp \omega_ 2\); (2) \(T_ 1\cap T_ 2\) has zero length (1-dimensional Hausdorff measure) where \(T_ j\) is the set of inner tangent points with respect to \(\Omega_ j\); \(j=1,2\).