Introduction

In this chapter we show that any volume preserving homeomorphism of the cube can be uniformly approximated by volume preserving automorphisms (not generally continuous) with certain specified measure theoretic properties. As shown in the previous section (when the property was ergodicity), this approximation can then be combined with the Lusin Theory to produce homeomorphisms possessing that property, and a version of Theorem C (of Chapter 1) for generic properties of volume preserving homeomorphisms of the cube.

Suppose we want to find homeomorphisms of In which have some particular measure theoretic property, such as ergodicity or weak mixing. Such a property can be designated by specifying a subset V of the space G[X, μ] of all automorphisms of a Lebesgue space (X, μ), which we will take for convenience as all volume preserving bijections of (In, λ). We will only consider properties V which don't depend on the names of the points, i.e., which are conjugate invariant in G[In, λ]. (This assumption means that g ∈ V implies f−1gf ∈ V for all f ∈ G[In, λ].) In this context the statement at the beginning of this paragraph is equivalent to showing V ∩ M[In, λ] is nonempty. Many important measure theoretic properties in G[In, λ] are determined by a countable number of conditions, which each define an open set in the weak topology on G[In, λ], that is, they are Gδ subsets of G[In, λ].