  We prove a measurable version of the Hall marriage theorem for actions of
finitely generated abelian groups. In particular, it implies that for free
measure-preserving actions of such groups, if two equidistributed measurable
sets are equidecomposable, then they are equidecomposable using measurable
pieces. The latter generalizes a recent result of Grabowski, M\'ath\'e and
Pikhurko on the measurable circle squaring and confirms a special case of a
conjecture of Gardner.
