Measurable equidecompositions for group actions with an expansion property

@article{Grabowski2022MeasurableEF,
  title={Measurable equidecompositions for group actions with an expansion property},
  author={Lukasz Grabowski and Andr'as M'ath'e and Oleg Pikhurko},
  journal={Journal of the European Mathematical Society},
  year={2022}
}
Given an action of a group Gamma on a measure space X, we provide a criterion under which two subsets A and B of X are measurably equidecomposable, i.e. A can be partitioned into finitely many measurable pieces, which can be rearranged using the elements of Gamma to form a partition of B. In particular, we show that every bounded measurable subset of R^n, n > 2, with non-empty interior is measurably equidecomposable to a ball. Similar result holds e.g. for measurable subsets of the unit sphere… 

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