Measurable equidecompositions via combinatorics and group theory
  We give a sketch of proof that any two (Lebesgue) measurable subsets of the unit sphere in $R^n$, for $n\ge 3$, with non-empty interiors and of the same measure are equidecomposable using pieces that are measurable. 
{\L}ukasz Grabowski, Andr\'as M\'ath\'e, Oleg Pikhurko
math.MG