Measurable equidecompositions via combinatorics and group theory

TL;DR

This paper sketches a proof that any two measurable subsets of the unit sphere in R^n with non-empty interiors and equal measure are measurably equidecomposable, combining combinatorics and group theory techniques.

Contribution

It introduces a novel approach to equidecomposition on spheres using combinatorics and group theory, extending previous results to measurable sets.

Findings

Measurable equidecomposition is possible for subsets of the sphere with equal measure.

The proof employs combinatorial and group-theoretic methods.

The approach applies to spheres in dimensions three and higher.

Abstract

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.