##
**Measure theory. Vol. 5. Set-theoretic measure theory. Part I, II.**
*(English)*
Zbl 1166.28002

Colchester: Torres Fremlin (ISBN 978-0-9538129-5-0/Pt.I; 978-0-9538129-6-7/Pt.II). 292 p. (2008).

The intentions of the author connected with the fifth volume (entitled: “Broad foundations”) of the treatise “Measure theory” is best illustrated by a quotation from the introduction: “For the final volume of this treatise, I have collected results which demand more sophisticated set theory than elsewhere. The line is not sharp, but typically we are much closer to questions which are undecidable in ZFC. Only in Chapter 55 are these brought to the forefront of the discussion, but elsewhere much of the work depends on formulations carefully chosen to express, as arguments in ZFC, ideas which arose in contexts in which some special axiom – Martin’s axiom, for instance – was being assumed”.

The first chapter “is centered on a study of partially ordered set”. In this part of the book the author talks over the following problems: ideals of sets, supported relations and Galois-Tukey connections, Stone spaces, cardinal functions of Boolean algebras, free subalgebras, Balcar-Frenek and Pierce-Koppelberg theorems, precalibers of supported relations, Martin numbers and Freese-Nation numbers of partially ordered sets.

The main ideas of the results contained in the chapter 52 are best conveyed by the author: “From the point of view of this book, the most important cardinals are those associated with measures and measure algebras, especially, of course, Lebesgue measure and the usual measure \(\mu_{I}\) of \(\{ 0,1\}^{I}\). In this chapter I try to cover the principal known facts about these which are theorems of ZFC. I start with a review of the theory for general measure spaces in §521, including some material which returns to the classification scheme of Chapter 21, exploring relationships between (strict) localizability, magnitude and Maharam type. §522 examines Lebesgue measure and the surprising connexions found by BARTOSZYNSKI 84 and RAISONNIER & STERN 85 between the cardinals associated with the Lebesgue null ideal and the corresponding ones based on the ideal of meager subsets of K. §523 looks at the measures \(\mu_{I}\) for uncountable sets \(I\), giving formulae for the additivities and cofinalities of their null ideals, and bounds for their covering numbers, uniformities and shrinking numbers. Remarkably, these cardinals are enough to tell us most of what we want to know concerning the cardinal functions of general Radon measures and semi-finite measure algebras (§524). These three sections are heavily dependent on the Galois-Tukey connections and Tukey functions of §§512-513. Precalibers do not seem to fit into this scheme, and the relatively partial information I have is in §525. The second half of the chapter deals with special topics which can be approached with the methods so far developed. In §526 I return to the ideal of subsets of \(\mathbb{N}\) with asymptotic density zero, seeking to locate it in the Tukey classification. Further \(\sigma\)-ideals which are of interest in measure theory are the ‘skew products’ of §527. In §528 I examine some interesting Boolean algebras, the ‘amoeba algebras’ first introduced by MARTIN & SOLOVAY 70, giving the results of TRUSS 88 on the connexions between different amoeba algebras and localization posets. Finally, in §529, I look at a handful of other structures, concentrating on results involving cardinals already described”.

In the Chapter 53 (entitled “Topologies and measures”) the author returns to problems considered in earlier volumes. §531 is connected with general Radon measures (Maharam types of Radon measures). Most notably, the author examines the class \(\text{Mah}_{R}(X)\) – the set of Maharam types of Maharam-type-homogeneous Radon probability measures on a Hausdorff space \(X\), while in §532 the author considers the class \(\text{Mah}_{cr}(X)\) – the set of Maharam types of Maharam-type-homogeneous completion regular topological probability measures on topological space \(X\). To describe the next parts of this Chapter let us quote the author: “In §534 I set out the elementary theory of ‘strong measure zero’ ideals in uniform spaces, concentrating on aspects which can be studied in terms of concepts already induced. Here there are some very natural questions which have not I think been answered (534Z). In the same section I run through the properties of Hausdorff measures when examined in the light of the concepts in Chapter 52. In §535 I look at liftings and strong liftings, extending the results of §§341 and 453; in particular, asking which non-complete probability spaces have liftings. In §536 I run over what is known about Alexandra Bellow’s problem concerning pointwise compact sets of continuous functions. With a little help from special axioms, there are some striking possibilities concerning repeated integrals, which I examine in §537. In §539, I complete my account of the result of B. Balcar, T. Jech and T. Pazak that it is consistent to suppose that every Dedekind complete ccc weakly \((\sigma ,\infty )\)-distributive Boolean algebra is a Maharam algebra, and work through applications of the methods of Chapter 52 to Maharam submeasures and algebras. Moving into new territory, I devote a section to a study of special types of filter on \(\mathbb{N}\) associated with measure-theoretic phenomena, and to medial limits”.

The Chapter 54 the author begins with the question: is there a non-trivial measure space in which every set is measurable? The first two paragraphs of this chapter have an introducing character and contain statements connected with: (\(\omega_{1}\)) saturated ideals and cardinal arithmetic. The main result contained in §543 is the Gitik-Shelah theorem: Let \(\kappa\) be an atomlessly-measurable cardinal, with witnessing probability \(\nu\), then the Maharam type of \(\nu\) is at least \(\min (\kappa^{(+\omega )},2^{\kappa})\).

The mainstream of problems of the paragraph 544 is well described in the introduction to this part of this book: “As is to be expected, a witnessing measure on a real-valued-measurable cardinal has some striking properties, especially if it is normal. What is less obvious is that the mere existence of such a cardinal can have implications for apparently unrelated questions in analysis. In 544J, for instance, we see that if there is any atomlessly-measurable cardinal then we have a version of Fubini’s theorem, \(\int \int f(x,y)dx dy = \int \int f(x,y)dy dx\), for many functions \(f\) on \(\mathbb{R}^{2}\) which are not jointly measurable. In this section I explore results of this kind. We find that, in the presence of an atomlessly-measurable cardinal, the covering number of the Lebesgue null ideal is large while its uniformity is small. There is a second inequality on repeated integrals to add to the one already given in 543C, and which tells us something about measure-precalibers; I add a couple of variations (544I-544J). Next, I give a pair of theorems on a measure-combinatorial property of the filter of conegligible sets of a normal witnessing measure. Revisiting the theory of Borel measures on metrizable spaces, discussed in §438 on the assumption that no real-valued-measurable cardinal was present, we find that there are some non-trivial arguments applicable to spaces with non-measure-free weight (544K-544L)”.

The fifth paragraph of this chapter is devoted to “product measure extension axiom” (PMEA) and “normal measure axiom” (NMA). The main idea of the paragraphs 546 and 547 is signaled in the introduction to these sections: “One way of interpreting the Gitik-Shelah theorem is to say that it shows that ‘simple’ atomless probability algebras cannot be of the form \({\mathcal P}X/{\mathcal N}(\mu)\). Similarly, the results of §541–§542 show that any ccc Boolean algebra expressible as the quotient of a power set by a non-trivial \(\sigma\)-ideal involves us in dramatic complexities, though it is not clear that these must appear in the quotient algebra itself. In this section I give two further results of M. Gitik and S. Shelah showing that certain algebras cannot appear in this way. I try to present the ideas in a form which leads naturally to some outstanding questions”. “Given a family \(\{ A_{i}\}_{i\in I}\) of sets in measure space, when can we find a disjoint family \(\{ A_{i}^{\prime}\}_{i\in I}\) such that \( A_{i}^{\prime} \subseteq A_{i}\) has the same outer measure as \( A_{i}\) for every \(i\)? A partial result is in Theorem 547F. Allied questions are: when can we find a set \(D\) such that \(A_{i}\cap D\) and \(A_{i} \setminus D\) have the same outer measure as \(A_{i}\) for every \(i\)? (547G) or are just non-negligible? (5471).”

The paragraphs 551–555 (the chapter 55 entitled: “Possible worlds”) are well exemplified in the introduction written by the author: “For a measure theorist, by far the most important forcings are those of ‘adding random reals’. I give three sections to these. Without great difficulty, we can determine the behaviour of the cardinals in Cichoń’s diagram, at least if many random reals are added. Going deeper, there are things to be said about outer measure and Sierpiński sets, and extensions of Radon measures. In the same section I give a version of the fundamental result that simple iteration of random real forcings gives random real forcings. In §553 I collect results which are connected with other topics dealt with above (Rothberger’s property, precalibers, ultrafilters, medial limits) and in which the arguments seem to me to develop properties of measure algebras which may be of independent interest. In preparation for this work, and also for §554, I start with a section devoted to a rather technical general account of forcings with quotients of \(\sigma\)-algebras of sets, aiming to find effective representations of names for points, sets, functions, measure algebras and filters.

Very similar ideas can also take us a long way with Cohen real forcing. Here I give little more than obvious parallels to the first part of §552, with an account of Freese-Nation numbers sufficient to support Carlson’s theorem that a Borel lifting of Lebesgue measure can exist when the continuum hypothesis is false.

One of the most remarkable applications of random reals is in Solovay’s proof that if it is consistent to suppose that there is a two-valued-measurable cardinal, then it is consistent to suppose that there is an atomlessly-measurable cardinal. By taking a bit of trouble over the lemmas, we can get a good deal more, including the corresponding theorem relating supercompact cardinals to the normal measure axiom; and similar techniques show the (possibility of interesting power set \(\sigma\)-quotient algebras”. In the first part of the paragraph 556 the author develops the topic “Forcing with Boolean algebra”. In the further part one can find some theorems with proofs connected with these methods: a strong law with large numbers; Dye’s theorem on orbit isomorphic measure-preserving transformations; Kawada theorem on invariant measure.

The last part of the book is entitled “Choice and determinancy”. In the first part of this chapter “Analysis without choice” the author “looks at basic facts from real analysis, functional analysis and general topology” which can be proved in Zermelo–Fraenkel set theory. The starting point of these considerations is a remark that in the absence of choice, the union of a sequence of countable sets need not be countable. For the description of remaining parts of this chapter we can quote the author: “§562 deals with ‘codable’ Borel sets and functions, using Borel codes to keep track of constructions for objects, so that if we know a sequence of codes we can avoid having to make a sequence of choices. A ‘Borel-coded measure’ is now one which behaves well with respect to codable sequences of measurable sets; for such a measure we have an integral with versions of the convergence theorems, and Lebesgue measure fits naturally into the structure. In §566, with ZF + AC(\(\omega\)), we are back in familiar territory, and most of the results of Volumes 1 and 2 can be proved if we are willing to re-examine some definitions and hypotheses. Finally, in §567, I look at infinite; games and half a dozen of the consequences of AD, with a postscript on determinacy in the context of ZF + AC.”

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 the set theory, Shelah’s pcf theory (reduced products, cofinalities); forcing; general topology (cardinal functions, Vietoris topology, category and Baire property, paracompact spaces), real-entiere functions.

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

The first chapter “is centered on a study of partially ordered set”. In this part of the book the author talks over the following problems: ideals of sets, supported relations and Galois-Tukey connections, Stone spaces, cardinal functions of Boolean algebras, free subalgebras, Balcar-Frenek and Pierce-Koppelberg theorems, precalibers of supported relations, Martin numbers and Freese-Nation numbers of partially ordered sets.

The main ideas of the results contained in the chapter 52 are best conveyed by the author: “From the point of view of this book, the most important cardinals are those associated with measures and measure algebras, especially, of course, Lebesgue measure and the usual measure \(\mu_{I}\) of \(\{ 0,1\}^{I}\). In this chapter I try to cover the principal known facts about these which are theorems of ZFC. I start with a review of the theory for general measure spaces in §521, including some material which returns to the classification scheme of Chapter 21, exploring relationships between (strict) localizability, magnitude and Maharam type. §522 examines Lebesgue measure and the surprising connexions found by BARTOSZYNSKI 84 and RAISONNIER & STERN 85 between the cardinals associated with the Lebesgue null ideal and the corresponding ones based on the ideal of meager subsets of K. §523 looks at the measures \(\mu_{I}\) for uncountable sets \(I\), giving formulae for the additivities and cofinalities of their null ideals, and bounds for their covering numbers, uniformities and shrinking numbers. Remarkably, these cardinals are enough to tell us most of what we want to know concerning the cardinal functions of general Radon measures and semi-finite measure algebras (§524). These three sections are heavily dependent on the Galois-Tukey connections and Tukey functions of §§512-513. Precalibers do not seem to fit into this scheme, and the relatively partial information I have is in §525. The second half of the chapter deals with special topics which can be approached with the methods so far developed. In §526 I return to the ideal of subsets of \(\mathbb{N}\) with asymptotic density zero, seeking to locate it in the Tukey classification. Further \(\sigma\)-ideals which are of interest in measure theory are the ‘skew products’ of §527. In §528 I examine some interesting Boolean algebras, the ‘amoeba algebras’ first introduced by MARTIN & SOLOVAY 70, giving the results of TRUSS 88 on the connexions between different amoeba algebras and localization posets. Finally, in §529, I look at a handful of other structures, concentrating on results involving cardinals already described”.

In the Chapter 53 (entitled “Topologies and measures”) the author returns to problems considered in earlier volumes. §531 is connected with general Radon measures (Maharam types of Radon measures). Most notably, the author examines the class \(\text{Mah}_{R}(X)\) – the set of Maharam types of Maharam-type-homogeneous Radon probability measures on a Hausdorff space \(X\), while in §532 the author considers the class \(\text{Mah}_{cr}(X)\) – the set of Maharam types of Maharam-type-homogeneous completion regular topological probability measures on topological space \(X\). To describe the next parts of this Chapter let us quote the author: “In §534 I set out the elementary theory of ‘strong measure zero’ ideals in uniform spaces, concentrating on aspects which can be studied in terms of concepts already induced. Here there are some very natural questions which have not I think been answered (534Z). In the same section I run through the properties of Hausdorff measures when examined in the light of the concepts in Chapter 52. In §535 I look at liftings and strong liftings, extending the results of §§341 and 453; in particular, asking which non-complete probability spaces have liftings. In §536 I run over what is known about Alexandra Bellow’s problem concerning pointwise compact sets of continuous functions. With a little help from special axioms, there are some striking possibilities concerning repeated integrals, which I examine in §537. In §539, I complete my account of the result of B. Balcar, T. Jech and T. Pazak that it is consistent to suppose that every Dedekind complete ccc weakly \((\sigma ,\infty )\)-distributive Boolean algebra is a Maharam algebra, and work through applications of the methods of Chapter 52 to Maharam submeasures and algebras. Moving into new territory, I devote a section to a study of special types of filter on \(\mathbb{N}\) associated with measure-theoretic phenomena, and to medial limits”.

The Chapter 54 the author begins with the question: is there a non-trivial measure space in which every set is measurable? The first two paragraphs of this chapter have an introducing character and contain statements connected with: (\(\omega_{1}\)) saturated ideals and cardinal arithmetic. The main result contained in §543 is the Gitik-Shelah theorem: Let \(\kappa\) be an atomlessly-measurable cardinal, with witnessing probability \(\nu\), then the Maharam type of \(\nu\) is at least \(\min (\kappa^{(+\omega )},2^{\kappa})\).

The mainstream of problems of the paragraph 544 is well described in the introduction to this part of this book: “As is to be expected, a witnessing measure on a real-valued-measurable cardinal has some striking properties, especially if it is normal. What is less obvious is that the mere existence of such a cardinal can have implications for apparently unrelated questions in analysis. In 544J, for instance, we see that if there is any atomlessly-measurable cardinal then we have a version of Fubini’s theorem, \(\int \int f(x,y)dx dy = \int \int f(x,y)dy dx\), for many functions \(f\) on \(\mathbb{R}^{2}\) which are not jointly measurable. In this section I explore results of this kind. We find that, in the presence of an atomlessly-measurable cardinal, the covering number of the Lebesgue null ideal is large while its uniformity is small. There is a second inequality on repeated integrals to add to the one already given in 543C, and which tells us something about measure-precalibers; I add a couple of variations (544I-544J). Next, I give a pair of theorems on a measure-combinatorial property of the filter of conegligible sets of a normal witnessing measure. Revisiting the theory of Borel measures on metrizable spaces, discussed in §438 on the assumption that no real-valued-measurable cardinal was present, we find that there are some non-trivial arguments applicable to spaces with non-measure-free weight (544K-544L)”.

The fifth paragraph of this chapter is devoted to “product measure extension axiom” (PMEA) and “normal measure axiom” (NMA). The main idea of the paragraphs 546 and 547 is signaled in the introduction to these sections: “One way of interpreting the Gitik-Shelah theorem is to say that it shows that ‘simple’ atomless probability algebras cannot be of the form \({\mathcal P}X/{\mathcal N}(\mu)\). Similarly, the results of §541–§542 show that any ccc Boolean algebra expressible as the quotient of a power set by a non-trivial \(\sigma\)-ideal involves us in dramatic complexities, though it is not clear that these must appear in the quotient algebra itself. In this section I give two further results of M. Gitik and S. Shelah showing that certain algebras cannot appear in this way. I try to present the ideas in a form which leads naturally to some outstanding questions”. “Given a family \(\{ A_{i}\}_{i\in I}\) of sets in measure space, when can we find a disjoint family \(\{ A_{i}^{\prime}\}_{i\in I}\) such that \( A_{i}^{\prime} \subseteq A_{i}\) has the same outer measure as \( A_{i}\) for every \(i\)? A partial result is in Theorem 547F. Allied questions are: when can we find a set \(D\) such that \(A_{i}\cap D\) and \(A_{i} \setminus D\) have the same outer measure as \(A_{i}\) for every \(i\)? (547G) or are just non-negligible? (5471).”

The paragraphs 551–555 (the chapter 55 entitled: “Possible worlds”) are well exemplified in the introduction written by the author: “For a measure theorist, by far the most important forcings are those of ‘adding random reals’. I give three sections to these. Without great difficulty, we can determine the behaviour of the cardinals in Cichoń’s diagram, at least if many random reals are added. Going deeper, there are things to be said about outer measure and Sierpiński sets, and extensions of Radon measures. In the same section I give a version of the fundamental result that simple iteration of random real forcings gives random real forcings. In §553 I collect results which are connected with other topics dealt with above (Rothberger’s property, precalibers, ultrafilters, medial limits) and in which the arguments seem to me to develop properties of measure algebras which may be of independent interest. In preparation for this work, and also for §554, I start with a section devoted to a rather technical general account of forcings with quotients of \(\sigma\)-algebras of sets, aiming to find effective representations of names for points, sets, functions, measure algebras and filters.

Very similar ideas can also take us a long way with Cohen real forcing. Here I give little more than obvious parallels to the first part of §552, with an account of Freese-Nation numbers sufficient to support Carlson’s theorem that a Borel lifting of Lebesgue measure can exist when the continuum hypothesis is false.

One of the most remarkable applications of random reals is in Solovay’s proof that if it is consistent to suppose that there is a two-valued-measurable cardinal, then it is consistent to suppose that there is an atomlessly-measurable cardinal. By taking a bit of trouble over the lemmas, we can get a good deal more, including the corresponding theorem relating supercompact cardinals to the normal measure axiom; and similar techniques show the (possibility of interesting power set \(\sigma\)-quotient algebras”. In the first part of the paragraph 556 the author develops the topic “Forcing with Boolean algebra”. In the further part one can find some theorems with proofs connected with these methods: a strong law with large numbers; Dye’s theorem on orbit isomorphic measure-preserving transformations; Kawada theorem on invariant measure.

The last part of the book is entitled “Choice and determinancy”. In the first part of this chapter “Analysis without choice” the author “looks at basic facts from real analysis, functional analysis and general topology” which can be proved in Zermelo–Fraenkel set theory. The starting point of these considerations is a remark that in the absence of choice, the union of a sequence of countable sets need not be countable. For the description of remaining parts of this chapter we can quote the author: “§562 deals with ‘codable’ Borel sets and functions, using Borel codes to keep track of constructions for objects, so that if we know a sequence of codes we can avoid having to make a sequence of choices. A ‘Borel-coded measure’ is now one which behaves well with respect to codable sequences of measurable sets; for such a measure we have an integral with versions of the convergence theorems, and Lebesgue measure fits naturally into the structure. In §566, with ZF + AC(\(\omega\)), we are back in familiar territory, and most of the results of Volumes 1 and 2 can be proved if we are willing to re-examine some definitions and hypotheses. Finally, in §567, I look at infinite; games and half a dozen of the consequences of AD, with a postscript on determinacy in the context of ZF + AC.”

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 the set theory, Shelah’s pcf theory (reduced products, cofinalities); forcing; general topology (cardinal functions, Vietoris topology, category and Baire property, paracompact spaces), real-entiere functions.

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 |

03-02 | Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations |