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.

The property of this chapter historically precedes that of absolute measurable spaces. The works of Sierpiński and Szpilrajn [142] and Szpilrajn–Marczewski [152] make more natural the introduction of absolute measurable spaces before the development of universally measurable sets in a space. The universally measurable property concerns sets in a fixed separable metrizable space rather than the property of topological embedding of a space into other spaces. This change of emphasis will be highlighted by switching the modifier “absolute” to “universally.” Interesting situations arise when the fixed space is absolute measurable.

The notion of a universally measurable set in a space is more complicated than that of absolute measurable spaces. Emphasis will be placed on the interplay between universally measurable sets in a space and absolute measurable subspaces. Of particular importance is the coinciding of universally null sets in a space X and the absolute null subspaces of X. Included is a presentation of a sharpening, due to Darst and Grzegorek, of the Purves theorem.

A closure–like operation, called the universally positive closure, is introduced to facilitate the study of the topological support of measures on X. This closure operation is used to define positive measures, those whose topological supports are as large as possible. It is shown that the notion of universally measurable sets in X can be achieved by using only those measures that are positive.

The Grzegorek and Ryll–Nardzewski solution to the natural question of symmetric differences of Borel sets and universally null sets is given.

Except for two statements in the earlier chapters that used the continuum hypothesis (abbreviated as CH), all the others used only what is now called the usual axioms of set theory – namely, the Zermelo–Frankel axioms plus the axiom of choice, ZFC for short. In this final chapter a look at the use of the continuum hypothesis and the Martin axiom in the context of absolute null space will be made. The discussion is not a thorough coverage of their use–the coverage is only part of the material that is found in the many references cited in the bibliography.

It has been mentioned many times that absolute null space is an example of the so–called singular sets. This example is a topological notion in the sense that it does not depend on the choice of a metric:F Two other metric independent singular sets will be included also. They are the Lusin set and the Sierpiński set in a given ambient space X.With regards to ambient spaces, it is known that “absolute null subspace of an ambient space X” is equivalent to “universally null set in X.”

The chapter is divided into four sections. The first is a rough historical perspective of the use of the continuum hypothesis in the context of universally null sets in a given space X. The second concerns cardinal numbers of absolute null spaces. The third is a brief discussion of the Martin axiom and its application to the above mentioned singular sets.

It is well–known that a compact metrizable space X is homeomorphic to {0, 1}ℕ if and only if X is nonempty, perfect and totally disconnected (hence, zero–dimensional). The classical Cantor ternary set in ℝ is one such, thus the name Cantor spaces. There are many other classical examples. A useful one is the product space kℕ, where k is a finite space endowed with the discrete topology and with card(k) > 1. It will be necessary that Cantor spaces be investigated not only as topological spaces but also as metric spaces with suitably assigned metrics.

The development presented in this appendix is based on E. Akin [2], R. Dougherty, R. D. Mauldin and A. Yingst [47], and O. Zindulka [162, 161]. There are two goals. The first is to present specific metrics on Cantor spaces which are used in the computations of Hausdorff measure and Hausdorff dimension in Chapter 5. The second is to discuss homeomorphic measures on Cantor spaces. The lack of an analogue of the Oxtoby–Ulam theorem for Cantor spaces motivates this goal.

Topologically characterizing homeomorphic, continuous, complete, finite Borel measures on Cantor spaces is a very complex task which has not been achieved yet. Simple topological invariants do not seem to characterize the homeomorphism classes of such measures. By introducing a linearly ordered topology consistent with the given topology of a Cantor space, which is always possible, a linear topological invariant has been discovered by Akin in [2].

In this chapter, attention is turned to topics in analysis such as measurability, derivatives and integrals of real–valued functions. Several connections between real–valued functions of a real variable and universally measurable sets in R have appeared in the literature. Four connections and their generalizations will be presented. The material developed in the earlier chapters are used in the generalizations. The fifth topic concerns the images of Lusin spaces under Borel measurable real valued functions – the classical result that these images are absolute null spaces will be proved. A brief description of the first four connections is given next before proceeding.

The first connection is a problem posed by A. J. Goldman [64] about σ–algebras associated with Lebesgue measurable functions; Darst's solution [35] will be given. A natural extension of Darst's theorem will follow from results of earlier chapters. Indeed, it will be shown that the domain of the function can be chosen to be any absolute measurable space that is not an absolute null space.

The second addresses the question of whether conditions such as bounded variation or infinitely differentiability have connections to theorems such as Purves's theorem; namely, for such functions, are the images of universally measurable sets in ℝ necessarily universally measurable sets in ℝ? Darst's negative resolutions of these questions will be presented.