5 - Hausdorff measure and dimension

Summary

There are two ways of looking at the dimension of a space–that is, topologically and measure theoretically. The measure theoretic dimension is the Hausdorff dimension, which is a metric notion. Hence, in this chapter, it will be necessary to assume that a metric has been or will be selected whenever the Hausdorff dimension is involved. The chapter concerns the Hausdorff measure and Hausdorff dimension of universally null sets in a metric space. The recent results of O. Zindulka [160, 161, 162, 163] form the major part of the chapter.

There are two well–known theorems [79, Chapter VII], which are stated next, that influence the development of this chapter.

Theorem 5.1. For every separable metric space, the topological dimension does not exceed the Hausdorff dimension.

Theorem 5.2. Every nonempty separable metrizable space has a metric such that the topological dimension and the Hausdorff dimension coincide.

The first theorem will be sharpened. Indeed, it will be shown that there is a universally null subset whose Hausdorff dimension is not smaller than the topological dimension of the metric space.

Universally null sets in metric spaces

We begin with a description of the development of Zindulka's theorems on the existence of universally null sets with large Hausdorff dimensions.

Zindulka's investigation of universally null sets in metric spaces begins with compact metrizable spaces that are zero–dimensional. The cardinality of such a space is at most ℵ₀ or exactly c. The first is not very interesting from a measure theoretic point of view.