##
**Measure theory. Vol. 4. Topological measure spaces. Part I, II.
Corrected second printing of the 2003 original.**
*(English)*
Zbl 1166.28001

Colchester: Torres Fremlin (ISBN 0-9538129-4-4). 528 p./Pt. I, 439 p., 19 p. (errata)/Pt. II. (2006).

The fourth volume of the treatise, entitled “Topological measure spaces” is devoted to study measure in the context of space and topology.

The first chapter of this volume is “an introduction to some of the most important ways in which topologies and measures can interact”. The starting point is to give some definitions (e.g. inner regular measure, topological measure space, \(\tau\)-additive measure, outer regular measure, quasi-Radon measure space, Borel and Baire measures, support set) and some examples. Successive paragraphs of this chapter develop the problems connected with notions introduced in the first paragraph.

The second chapter is some introduction to the descriptive set theory. The intention of the author connected with this part of the treatise is best illustrated by a quotation: “The first section describes Souslin’s operation and its basic set-theoretic properties up to the theory of ‘constituents’, mostly steering away from topological ideas, but with some remarks on \(\sigma\)-algebras and Souslin-F sets. §422 deals with usco-compact relations and K-analytic spaces, working through the topological properties which will be useful later, and giving a version of the First Separation Theorem. §423 looks at ‘analytic’ or ‘Souslin’ spaces, treating them primarily as a special kind of K-analytic space, with the von Neumann-Jankow selection theorem. §424 is devoted to ‘standard Borel spaces’; it is largely a series of easy applications of results in §423, but there is one substantial theorem on Borel measurable actions of Polish groups.”

The chapter with number 43 (third chapter in the fourth volume) is some continuation of considerations contained in the first chapter of this volume. The paragraph begins with the section concerning Souslin operations. In the next section the author describes basic measure-theoretic properties of K-analytic spaces. The main result of this section is the theorem (Aldaz, Render):

Let \(X\) be a K-analytic Hausdorff space and \(\mu\) a locally finite measure on \(X\) which is inner regular with respect to the closed sets. Then \(\mu\) has an extension to a Radon measure on \(X\) (in particular \(\mu\) is \(\tau\) additive).

In the third paragraph the author considers some properties of analytic spaces. The fourth paragraph the author begins with the formulation of the basic question: “What kinds of measures can arise on what kinds of topological space?” The main idea of this paragraph (entitled “Borel measures”) is presented in some parts of introduction: “In 434A I set out a crude classification of Borel measures on topological spaces. For compact Hausdorff spaces, at least, the first question is whether they carry Borel measures which are not, in effect, Radon measures; this leads us to the definition of ‘Radon’ space which is also of interest in the context of general Hausdorff spaces. I give a brief account of the properties of Radon spaces. I look also at two special topics: ‘quasi-dyadic’ spaces and a construction of Borel product measures by integration of sections. In the study of Radon spaces we find ourselves looking at ‘universally measurable’ subsets of topological spaces. […] Three further classes of topological space, defined in terms of the types of topological measure which they carry, are the ‘Borel-measure-compact’, ‘Borel-measure-complete’ and ‘pre-Radon’ spaces; I discuss them briefly in 434G-434J. They provide useful methods for deciding whether Hausdorff spaces are Radon”.

The fourth chapter the author begins with the paragraph containing theorem on the existence of invariant measures: a locally compact Hausdorff topological group has left and right Haar measures, which are both Radon measures. The next two paragraphs describe properties and emphasize the uniqueness of Haar measures. Paragraph 444 is devoted to convolution measures, function and measures and, moreover, contains general results concerning continuous group actions on quasi-Radon measure spaces. The successive part of this chapter contains a proof of the Pontryagin-van Kampen duality theorem. In the last four paragraphs the author tackles the following problems: the structure of locally compact groups, translation-invariant liftings and lower densities; Vitali’s theorem and a density theorem for groups with \(B\)-sequences; invariant measures on Polish spaces and (locally compact) amenable groups.

The chapter 45 (the last chapter in the first part of the fourth volume) is entitled “Perfect measures and disintegration”. The starting points of the considerations are the notions: compact measure and countably compact measures and the Ryll-Nardzewski theorem: Any semi-finite countably compact measure is perfect. §452 is devoted to the problem of integration and disintegration of measures (special types of disintegration connected with the phrase: “regular conditional probability”). In the next paragraph the author describes some cases in which strong liftings are known to exist. This part of the chapter is finished with a note on the relation between strong liftings and a Stone space and with an example of a space with no strong lifting. Paragraph 454 (entitled: “Measures on product spaces”) contains, among other things, the Marczewski and Ryll-Nardzewski Thorem. The next part is connected with Markov processes, which lead to the straightforward existence theorem “dependent only on a natural consistency condition on the conditional distributions”. In the last parts of this chapter the author gives results connected with (universal) Gaussian distributions, extensions of measures (Strassen’s theorem); relatively independent families of \(\sigma\) algebras and random variables; relative distribution; relative products of probability spaces; exchangeable families of inverse-measure-preserving functions and symmetric quasi-Radon measures.

The second part of the fourth volume (chapter 46) begins with the theme: “Pointwise compact sets of measurable functions”. Within the range of this theme the following barycenters (sufficient conditions for existence, Krein theorem, measures on sets of extreme points); pointwise compact sets of continuous functions (the topology of pointwise convergence on \(C(X)\), weak convergence, convex hulls, separately continuous function); pointwise convergence on spaces of measurable functions; Talangrand’s measure; stable sets of functions; quasi-Radon measures for weak and strong topologies; universally measurable linear operators and locally uniformly rotund norms.

In the introduction to the Chapter 47 (entitled: “Geometric measure theory”) the author writes: “The greater part of it is directed specifically at a version of the Divergence Theorem”. The first paragraph is devoted to Hausdorff measures on general metric spaces. The further parts of this chapter the author describes in the following way: “§472, at least, deals with something which must be central to any approach, Besicovitch’s Density Theorem for Radon measures on \(\mathbb{R}^{r}\). In §473 I examine Lipschitz functions, and give crude forms of some fundamental inequalities relating integrals \(\int \| \text{grad} f \|d\mu\) with other measures of the variation of a function \(f\). In §474 I introduce perimeter measures \(\lambda^{\partial}_{E}\) and outward-normal functions \(\psi_{E}\) as those for which the Divergence Theorem, in the form \(\int_{E}\text{div}\phi d\mu =\int \phi \cdot \psi d\lambda^{\partial}_{E} \) will be valid, and give the geometric description of \(\psi_{e}(x)\) as the Federer exterior normal to \(E\) at \(x\). In §475 I show that \(\lambda^{\partial}_{E}\) can be identified with normalized Hausdorff (\(r-l\))-dimensional measure on the essential boundary of \(E\). […] In §476 I turn to a different topic, the problem of finding the subsets of \(\mathbb{R}^{r}\) on which Lebesgue measure is most ‘concentrated’ in some sense. I present a number of classical results, the deepest being the Isoperimetric Theorem: among sets with a given measure, those with the smallest perimeters are the balls.”

The Chapter 48 (entitled “Gauge integrals”) is devoted to the Kurzweil - Henstock integrals. The first paragraph contains the terminology and the description of examples. The best description of the content of next chapters gives the author in the introduction to this part of the treatise: “In §482 I give a handful of general theorems showing what kinds of result can be expected and what difficulties arise. In §483, I work through the principal properties of the Henstock integral on the real line, showing, in particular, that it coincides with the Perron and special Denjoy integrals. Finally, in §484, I look at a very striking integral on \(\mathbb{R}^{r}\), due to W. F. Pfeffer.”

The title of the last chapter: “Further topics” explains the intentions of the author. This part of the book contains the following problems: concentrations of measure in product space and in parmutations group; extremely amenable groups; locally compact groups; product sets included in given sets of positive measure; Poison distributions and Poisson point process.

This volume ends with appendices containing some information relevant to some topics presented in this volume. In this part of the treatise the author gives some facts connected with set theory and general topology, topological \(\sigma\) algebras; locally convex spaces; topological groups and Banach algebras.

Each section of this treatise ends with “basic exercises”, “further exercises” and “Notes and comments”.

The first chapter of this volume is “an introduction to some of the most important ways in which topologies and measures can interact”. The starting point is to give some definitions (e.g. inner regular measure, topological measure space, \(\tau\)-additive measure, outer regular measure, quasi-Radon measure space, Borel and Baire measures, support set) and some examples. Successive paragraphs of this chapter develop the problems connected with notions introduced in the first paragraph.

The second chapter is some introduction to the descriptive set theory. The intention of the author connected with this part of the treatise is best illustrated by a quotation: “The first section describes Souslin’s operation and its basic set-theoretic properties up to the theory of ‘constituents’, mostly steering away from topological ideas, but with some remarks on \(\sigma\)-algebras and Souslin-F sets. §422 deals with usco-compact relations and K-analytic spaces, working through the topological properties which will be useful later, and giving a version of the First Separation Theorem. §423 looks at ‘analytic’ or ‘Souslin’ spaces, treating them primarily as a special kind of K-analytic space, with the von Neumann-Jankow selection theorem. §424 is devoted to ‘standard Borel spaces’; it is largely a series of easy applications of results in §423, but there is one substantial theorem on Borel measurable actions of Polish groups.”

The chapter with number 43 (third chapter in the fourth volume) is some continuation of considerations contained in the first chapter of this volume. The paragraph begins with the section concerning Souslin operations. In the next section the author describes basic measure-theoretic properties of K-analytic spaces. The main result of this section is the theorem (Aldaz, Render):

Let \(X\) be a K-analytic Hausdorff space and \(\mu\) a locally finite measure on \(X\) which is inner regular with respect to the closed sets. Then \(\mu\) has an extension to a Radon measure on \(X\) (in particular \(\mu\) is \(\tau\) additive).

In the third paragraph the author considers some properties of analytic spaces. The fourth paragraph the author begins with the formulation of the basic question: “What kinds of measures can arise on what kinds of topological space?” The main idea of this paragraph (entitled “Borel measures”) is presented in some parts of introduction: “In 434A I set out a crude classification of Borel measures on topological spaces. For compact Hausdorff spaces, at least, the first question is whether they carry Borel measures which are not, in effect, Radon measures; this leads us to the definition of ‘Radon’ space which is also of interest in the context of general Hausdorff spaces. I give a brief account of the properties of Radon spaces. I look also at two special topics: ‘quasi-dyadic’ spaces and a construction of Borel product measures by integration of sections. In the study of Radon spaces we find ourselves looking at ‘universally measurable’ subsets of topological spaces. […] Three further classes of topological space, defined in terms of the types of topological measure which they carry, are the ‘Borel-measure-compact’, ‘Borel-measure-complete’ and ‘pre-Radon’ spaces; I discuss them briefly in 434G-434J. They provide useful methods for deciding whether Hausdorff spaces are Radon”.

The fourth chapter the author begins with the paragraph containing theorem on the existence of invariant measures: a locally compact Hausdorff topological group has left and right Haar measures, which are both Radon measures. The next two paragraphs describe properties and emphasize the uniqueness of Haar measures. Paragraph 444 is devoted to convolution measures, function and measures and, moreover, contains general results concerning continuous group actions on quasi-Radon measure spaces. The successive part of this chapter contains a proof of the Pontryagin-van Kampen duality theorem. In the last four paragraphs the author tackles the following problems: the structure of locally compact groups, translation-invariant liftings and lower densities; Vitali’s theorem and a density theorem for groups with \(B\)-sequences; invariant measures on Polish spaces and (locally compact) amenable groups.

The chapter 45 (the last chapter in the first part of the fourth volume) is entitled “Perfect measures and disintegration”. The starting points of the considerations are the notions: compact measure and countably compact measures and the Ryll-Nardzewski theorem: Any semi-finite countably compact measure is perfect. §452 is devoted to the problem of integration and disintegration of measures (special types of disintegration connected with the phrase: “regular conditional probability”). In the next paragraph the author describes some cases in which strong liftings are known to exist. This part of the chapter is finished with a note on the relation between strong liftings and a Stone space and with an example of a space with no strong lifting. Paragraph 454 (entitled: “Measures on product spaces”) contains, among other things, the Marczewski and Ryll-Nardzewski Thorem. The next part is connected with Markov processes, which lead to the straightforward existence theorem “dependent only on a natural consistency condition on the conditional distributions”. In the last parts of this chapter the author gives results connected with (universal) Gaussian distributions, extensions of measures (Strassen’s theorem); relatively independent families of \(\sigma\) algebras and random variables; relative distribution; relative products of probability spaces; exchangeable families of inverse-measure-preserving functions and symmetric quasi-Radon measures.

The second part of the fourth volume (chapter 46) begins with the theme: “Pointwise compact sets of measurable functions”. Within the range of this theme the following barycenters (sufficient conditions for existence, Krein theorem, measures on sets of extreme points); pointwise compact sets of continuous functions (the topology of pointwise convergence on \(C(X)\), weak convergence, convex hulls, separately continuous function); pointwise convergence on spaces of measurable functions; Talangrand’s measure; stable sets of functions; quasi-Radon measures for weak and strong topologies; universally measurable linear operators and locally uniformly rotund norms.

In the introduction to the Chapter 47 (entitled: “Geometric measure theory”) the author writes: “The greater part of it is directed specifically at a version of the Divergence Theorem”. The first paragraph is devoted to Hausdorff measures on general metric spaces. The further parts of this chapter the author describes in the following way: “§472, at least, deals with something which must be central to any approach, Besicovitch’s Density Theorem for Radon measures on \(\mathbb{R}^{r}\). In §473 I examine Lipschitz functions, and give crude forms of some fundamental inequalities relating integrals \(\int \| \text{grad} f \|d\mu\) with other measures of the variation of a function \(f\). In §474 I introduce perimeter measures \(\lambda^{\partial}_{E}\) and outward-normal functions \(\psi_{E}\) as those for which the Divergence Theorem, in the form \(\int_{E}\text{div}\phi d\mu =\int \phi \cdot \psi d\lambda^{\partial}_{E} \) will be valid, and give the geometric description of \(\psi_{e}(x)\) as the Federer exterior normal to \(E\) at \(x\). In §475 I show that \(\lambda^{\partial}_{E}\) can be identified with normalized Hausdorff (\(r-l\))-dimensional measure on the essential boundary of \(E\). […] In §476 I turn to a different topic, the problem of finding the subsets of \(\mathbb{R}^{r}\) on which Lebesgue measure is most ‘concentrated’ in some sense. I present a number of classical results, the deepest being the Isoperimetric Theorem: among sets with a given measure, those with the smallest perimeters are the balls.”

The Chapter 48 (entitled “Gauge integrals”) is devoted to the Kurzweil - Henstock integrals. The first paragraph contains the terminology and the description of examples. The best description of the content of next chapters gives the author in the introduction to this part of the treatise: “In §482 I give a handful of general theorems showing what kinds of result can be expected and what difficulties arise. In §483, I work through the principal properties of the Henstock integral on the real line, showing, in particular, that it coincides with the Perron and special Denjoy integrals. Finally, in §484, I look at a very striking integral on \(\mathbb{R}^{r}\), due to W. F. Pfeffer.”

The title of the last chapter: “Further topics” explains the intentions of the author. This part of the book contains the following problems: concentrations of measure in product space and in parmutations group; extremely amenable groups; locally compact groups; product sets included in given sets of positive measure; Poison distributions and Poisson point process.

This volume ends with appendices containing some information relevant to some topics presented in this volume. In this part of the treatise the author gives some facts connected with set theory and general topology, topological \(\sigma\) algebras; locally convex spaces; topological groups and Banach algebras.

Each section of this treatise ends with “basic exercises”, “further exercises” and “Notes and comments”.

Reviewer: Ryszard Pawlak (Łódź)

### MSC:

28-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to measure and integration |

26-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to real functions |

54-02 | Research exposition (monographs, survey articles) pertaining to general topology |