Cauchy transforms and characteristic functions

The following problem arises in the study of rational approximation: classify all plane sets E such that (z) = d ()l( -z) = E (z) area almost everywhere for some com- plex Borel measure . A partial solution to this problem for compact sets is given here. The main result is the following. THEOREM. Let K be a compact plane set with connected dense interior. Then there is a measure such that = area a.e., if and only if K has finite Painleve length.