Moment problems

The classical moment problem of Hausdorff is relevant in probability theory or spectral problems of selfadjoint operators. An open problem: Given an arbitrary Borel measure on the square X=[0,1] x [0,1]. Assume you know the expectation values of (x,y) -> x n y m (= moments) How does one decide from these moments, whether the measure is absolutely continuous? How can one decide whether mutual absolutely continuity holds for two given measures, if their moments are known?

Oliver home