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

Measurable Hall's theorem for actions of abelian groups