  If $\mu$ is a finite complex measure in the complex plane $\C$ we denote by
$C^\mu$ its Cauchy integral defined in the sense of principal value. The
measure $\mu$ is called reflectionless if it is continuous (has no atoms) and
$C^\mu=0$ at $\mu$-almost every point. We show that if $\mu$ is reflectionless
and its Cauchy maximal function $C^\mu_*$ is summable with respect to $|\mu|$
then $\mu$ is trivial. An example of a reflectionless measure whose maximal
function belongs to the "weak" $L^1$ is also constructed, proving that the
above result is sharp in its scale. We also give a partial geometric
description of the set of reflectionless measures on the line and discuss
connections of our results with the notion of sets of finite perimeter in the
sense of De Giorgi.
