Chapter 3 - Paradoxes in Measure Theory

Abstract

This chapter describes the importance of paradoxes in measure theory. The paradoxes are important because of their close connection to invariant measures. The simplest geometric form of the paradox of infinity is exhibited by any set that is congruent to a proper subset of itself. Every half-line has this property. A stronger form of the paradox of infinity states that an infinite set can be decomposed into two subsets, each of which is equivalent to the original. It is observed that equidecomposability of sets can be formulated in terms of perfect matchings of bipartite graphs. The connection between equidecomposability and perfect matchings is also explained in the chapter. The group of rotations of a sphere is locally commutative, and if two rotations have a common fixed point then they have the same axis and they commute. The countable equidecomposability and countably additive invariant measures are analyzed by different theorems. The nonconstructive element in the paradoxes is also elaborated in the chapter.