The measurable Tarski circle squaring problem

Two subsets A and B of R^n are equidecomposable if it is possible to partition A into pieces and rearrange them via isometries to form a partition of B. Motivated by what is nowadays known as Banach-Tarski paradox, Tarski asked if the unit square and the disc of unit area in R^2 are equidecomposable. 65 years later Laczkovich showed that they are, at least when the pieces are allowed to be non-measurable sets. I will talk about a joint work with A. Mathe and O. Pikhurko which implies in particular the existence of a measurable equidecomposition of circle and square in R^2.