  We construct mesures supported on a compact subset E of the real line having
zero principal value of their Cauchy integral a.e. on E with respect to
Lebesgue measure and having singular components. E is sufficiently regular
(Widom property is satisfied) but not homogeneous as for homogeneous spectrum
such construction is impossible. This impossibility played an important role in
characterizing almost periodic Jacobi matrices with homogeneous spectrum
(Sodin-Yuditskii).
