In what sense a function and its Cauhy transform can have disjoint support? What are compactly supported measures on the plane whose Cauchy transform vanishes a.e. with respect to this measure? These questions concern the so called reflectionless measures which play an important part for the theory of Schroedinger operators with ergodic potential and almost periodic Jacobi matrices. We present the information we know about such measures. We will meet a bit of Geometric Measure Theory and harmonic measure estimates in this talk.