Introduction

The Lusin Theorem (or rather its consequence Corollary 6.3) in the previous chapter provides us with a method of constructing volume preserving homeomorphisms with desired measure theoretic properties. This method reduces the problem to approximating a volume preserving homeomorphism uniformly by a volume preserving automorphism (not necessarily continuous) with the desired measure theoretic property. In the next chapter we will give a very general application of this method, but here we use it simply to demonstrate the existence (and typicality) of ergodic homeomorphisms of the cube. (We recall that an automorphism of a finite measure space is said to be ergodic if its only invariant sets are of measure zero or full measure.) Again, this is an optional chapter, in that a stronger result (Theorem 8.2) will be proved independently in the next chapter.

However, the proof we present here, that ergodicity is typical among volume preserving homeomorphisms of the cube, is a very clear illustration of the method of approximation by discontinuous automorphisms. Given Corollary 6.3 of the previous chapter, we are required only to approximate an arbitrary homeomorphism in M[In, λ] by an ergodic (generally discontinuous) automorphism in G[In, λ], in the uniform topology.

Theorem 7.1The ergodic homeomorphisms form a dense Gδ subset of the volume preserving homeomorphisms of In, in the uniform topology.

Proof Let G[In, λ] denote the space of all volume (λ) preserving bimeasurable bijections of the unit cube, endowed with the weak topology.