Introduction

A central idea of real variable theory, ‘Littlewood's Second Principle’, is that every measurable function is nearly continuous. Two forms of this principle are contained in the following well known result, the stronger second part of which is known as ‘Lusin's Theorem’.

Theorem 6.1Let g : R → R be a measurable real valued function with |g(x) – x| < ∈ on the interval [a,b]. Then for any δ > 0 there is a continuous function h : R → R with |h(x) – x| < ∈ on [a,b] satisfying

1. λ {x : | g(x)− h(x)| ≥ δ} < δ, and even

2. λ {x : g(x)≠ h(x)} < δ.

In this chapter we will prove an analogous result which relates measurable and continuous ergodic theory. That is, we show that a volume preserving bimeasurable bijection of the cube In is nearly a volume preserving homeomorphism. The notion of ‘nearly’ is made precise in the following result obtained by Alpern [8].

Theorem 6.2 (Measure Preserving Lusin Theorem)Let g be a bimeasurable volume preserving bijection (i.e., automorphism) of the cube In, n ≥ 2, with ∥g∥ ≡ ess sup |g(x)−x| < ∈. Then given any δ > 0, there is a volume preserving homeomorphism h of In, with ∥h∥ < ∈ and equal to the identity on the boundary of In, satisfying

1. λ {x : |g(x)−h(x)| ≥ δ} < δ

2. λ {x : g(x) ≠ h(x)} < δ.