Introduction to Part II

Up to now we have restricted our attention to volume preserving homeomorphisms of the cube, and have proved a number of results for this space M[In, λ]. In this part of the book (Chapters 9 and 10) we show how the results already obtained for M[In, λ] apply more generally to the space M[X, μ] whenever X is any compact connected manifold (we allow situations where our manifold X could possibly have nonempty boundary as for example when X = In) and μ belongs to a certain class of finite measures. In other words, we will show that there was really no loss of generality in restricting our attention to the cube with volume measure, where the intuition was clearer.

We note for later purposes that the situation is very different for noncompact manifolds, in that results obtained for the ‘standard noncompact manifold’ Rn do not go over unchanged to arbitrary noncompact manifolds. That is, for compact manifolds the topological type of the manifold is irrelevant, but for noncompact manifolds the end structure is important. But these are matters to be dealt with in Part III.

General Measures on the Cube

We begin our analysis by retaining for the moment the cube In, n ≥ 2, as our manifold, but now endowing it with a more general Borel probability measure μ.