# Measurable Hall's theorem for actions of abelian groups

## TL;DR

This paper establishes a measurable analogue of Hall's marriage theorem for finitely generated abelian group actions, showing that equidecomposable measurable sets can be decomposed into measurable pieces, extending previous results on circle squaring.

## Contribution

It introduces a measurable version of Hall's theorem for abelian group actions, advancing the understanding of measurable equidecomposability.

## Key findings

- Measurable Hall's theorem for abelian groups proved
- Equidecomposability implies measurable equidecomposability for free actions
- Generalizes results on measurable circle squaring

## Abstract

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.