Only separable metrizable spaces will be considered. Appendix A will be used to gather various notions and facts that are found in well–known reference books (for example, K. Kuratowski [85]) with the goal of setting consistent notation and easing the citing of facts. Also, the final Section A.7 will be used to present a proof, which includes a strengthening due to R. B. Darst [37], of a theorem of R. Purves [129].

A.1. Complete metric spaces

A metric that yields the topology for a metrizable space need not be complete. But this metric space can be densely metrically embedded into another metric space that is complete. A space that possesses a complete metric will be called completely metrizable. It is well–known that a Gδ subspace of a completely metrizable space is also completely metrizable (see J. M. Aarts and T. Nishiura [1, page 29]). Consequently,

Theorem A.1. A separable metrizable space is completely metrizable if and only if it is homeomorphic to a Gδ subset of a separable completely metrizable space.

The collection of all separable metrizable spaces will be denoted by MET and the collection of all completely metrizable spaces in MET will be denoted by METcomp.

Extension of a homeomorphism. In the development of absolute notions it is often useful to extend a homeomorphism between two subsets of completely metrizable spaces to a homeomorphism between some pair of δ subsets containing the original subsets. This is accomplished by means of the M. Lavrentieff theorem [89].

This appendix contains a summary of the needed topological dimension theory, and, for metric spaces, the needed Hausdorff measure theory and the Hausdorff dimension theory.

Topological dimension

There are three distinct dimension functions in general topology, two of which are inductively defined and the third is defined by means of open coverings. Each definition has its advantages and its disadvantages. Fortunately, the three agree whenever the spaces are separable and metrizable. Let us give their definitions.

Definition D.1. Let X be a topological space.

The space X is said to have small inductive dimension −1if and only if it is the empty space. For each positive integer n, the space X is said to have small inductive dimension not exceeding n if each point of X has arbitrarily small neighborhoods whose boundaries have small inductive dimension not exceeding n − 1. These conditions are denoted by ind X ≤ n. The definition of ind X = n is made in the obvious manner for n = −1, 0, 1,…, +∞

The space X is said to have large inductive dimension −1if and only if it is the empty space. For each positive integer n, the space X is said to have large inductive dimension not exceeding n if each closed subset of X has arbitrarily small neighborhoods whose boundaries have large inductive dimension not exceeding n−1. These conditions are denoted by Ind X ≤ n.

A measure space M(X, μ) is a triple (X, μ, (X, μ), where μ is a countably additive, nonnegative, extended real–valued function whose domain is the σ–algebra (X, μ) of subsets of a set X and satisfies the usual requirements. A subset M of X is said to be μ–measurable if M is a member of the μ–algebra M(X, μ).

For a separable metrizable space X, denote the collection of all Borel sets of X by B(X). A measure space M(X, μ) is said to be Borel if B(X) ⊂ M(X, μ), and if M ∈ M(X, μ) then there is a Borel set B of X such that M ⊂ B and μ(B) = μ(M)1. Note that if μ(M) < ∞, then there are Borel sets A and B of X such that A ⊂ M ⊂ B and μ(B \ A) = 0.

Certain collections of measure spaces will be referred to often – for convenience, two of them will be defined now.

Notation 1.1 (MEAS ; MEASfinite). The collection of all complete, σ–finite Borel measure spaces M(X, μ) on all separable metrizable spaces X will be denoted by MEAS. The subcollection of MEAS consisting of all such measures that are finite will be denoted by MEASfinite.

This book is about absolute measurable spaces. What is an absolute measurable space and why study them?

To answer the first question, an absolute measurable space, simply put, is a separable metrizable space X with the property that every topological embedding of X into any separable metrizable space Y results in a set that is μ–measurable for every continuous, complete, finite Borel measure μ on Y. Of course, only Borel measures are considered since the topology of Y must play a role in the definition.

For an answer to the second question, observe that the notion of absolute measurable space is a topological one in the spirit of many other notions of “absolute” such as absolute Borel space, absolute Gδ space, absolute retract and many more. As the definition is topological, one is led to many topological questions about such spaces. Even more there are many possible geometric questions about such spaces upon assigning a metric to the space. Obviously, there is also a notion of “absolute null space”; these spaces are those absolute measurable spaces for which all topological copies have μ measure equal to 0. Absolute null spaces are often called “universal measure zero sets” and have been extensively studied. The same topological and geometric questions can be investigated for absolute null spaces. It is well–known that absolute Borel spaces are absolute measurable spaces. More generally, so are analytic and co–analytic spaces.

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.