Equidescomponibilidad, el problema de la medida y la paradoja de Banach-Tarski

Abstract

In this work we study the problem of equidecomposability. Two sets are said to be equidecomposable if there is a finite family of pairwise disjoint subsets of one of them that can be reassembled by rigid motions to form the other. In the first section we prove two theorems on equidecomposability on the plane. The first result, Bolyai-Gerwien-Wallace theorem, states that two polygons with the same area are equidecomposable using only triangle pieces. The second theorem shows that the circle and the square are not equidecomposable only using cutout pieces (Jordan domains with piecewise differentiable boundary). In the second section we study this problem in the three dimensional space, proving the Banach-Tarski Paradox. On our way to this objective we need to prove Hausdorff’s Paradox as it sets the entry point of Banach-Tarski’s result. The BanachTarski Paradox establishes that any solid sphere in R 3 is paradoxical, meaning it can be divided into two disjoint subsets, each of them equidecomposable with the original sphere. Naturally, since Lebesgue measure is rigid motion invariant, the pieces considered in these paradoxes are non-measurable. In the last section we will return to the equidecomposability problem on lower dimensions to show no bounded subset of the real line and the plane with non-empty interior is paradoxical. This is known as Banach Theorem and it shows that finitely additive, invariant under isometries, extensions of the Lebesgue length or area measure exists, defined on all P(R) and P(R 2 ).