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.