Measurable equidecompositions Seminar

Two subsets A and B of R^3 are equidecomposable if it is possible to find a partition of A into finitely many pieces and rearrange these pieces using isometries to form a partition of B. Banach and Tarski showed that any two bounded sets with non-empty interiors in R^3 are equidecomposable. In opposition to the famous Banach-Tarski paradox, when A and B are of the same measure, there is no a priori reason why there should be no equidecomposition between A and B with measurable pieces. In the talk I'll describe a joint work with A. Mathe and O. Pikhurko where we show, under suitable assumptions on A and B, the existence of measurable equidecompositions. Our results imply in particular the measurable version of Hilbert's third problem: regular tetrahedron can be "cut" into measurable pieces, and the pieces rearranged to form a cube.