Measure theory. Vol. 2. Broad foundations.
Corrected second printing of the 2001 original.

*(English)*Zbl 1165.28001
Colchester: Torres Fremlin (ISBN 0-9538129-2-8/pbk). 563 p., 12 p. (errata). (2003).

The intentions of the author connected with the second volume (entitled: “Broad foundations”) of the treatise “Measure theory” is best illustrated by a quotation from the introduction: For this second volume I have chosen seven topics through which to explore the insights and challenges offered by measure theory. Some, like the Radon-Nikodým theorem (Chapter 23) are necessary for any understanding of the structure of the subject; others, like Fourier analysis (Chapter 28) and the discussion of function spaces (Chapter 24) demonstrate the power of measure theory to attack problems in general real and functional analysis. Writing “seven topics” the author omits the first part: 21. Taxonomy of measure spaces: because of ”It is the purest of pure measure theory”.

First section of Chapter 21 contains a list of definitions distinguishing different types of measure spaces. In the next sections the author gives a brief account of the theory of some measure spaces described in the first section. Part 214 is connected with subspaces, which lead to the ”integration over a subset”. In the last section of this chapter the author presents some examples which allow to establish relationships between various kinds of measure spaces. The starting point of the second chapter (marked number 22) is Vitali’s theorem in \(\mathbb{R}\). The main aim of the second section of this chapter is Theorem 222E connected with differentiation of indefinite integral. Next the author proceeds to the Lebesgue’s density theorems: first in the integral form (also for complex-valued functions) and in exercises in form leading to the density topology. The last three parts of this chapter contain the most important statements concerning the functions of bounded variation and absolutely continuous functions (as an example of applications of the results: Theorem of integration by parts, theorems for lower semi-continmuous functions and the Lebesgue decomposition of a function of bounded variation).

Chapter 23 is devoted to the Radon-Nikodým theorem. The first section of this chapter introduces the basic tools: countable additive functional and its decompositions. The main theorem is contained in the second paragraph (marked number 232), which is finished with considerations concerning Lebesgue decomposition of a countably additive functional. In the next part the author gives some application of the Radon-Nikodým theorem providing the concept of “conditional expectations”. The successive sections are devoted to the following problems: indefinite-integral measures from a non-negative measurable function on a measure space; general method of integration by substitution; inverse measure preserving functions.

In Chapter 24 the author discusses function spaces \(L^{0}\), \(L^{p}\) and \(L^{\infty}\). In this field the following problems are analyzed more minutely: Riesz spaces, the space \(L^{1}\) (the norm of \(L^{1}\)), a dense subspaces of \(L^{\infty}\); norms of \(L^{p}\); convergence in measure; pointwise convergence; embedding \(L^{p}\) in \(L^{0}\); uniform integrability; compactness in \(L^{1}\). In paragraph 246Y (“Further exercise”) the Viltali-Hahn-Saks theorem is contained).

In Chapter 25 the author considered product measure: two (finite) product (§251) and the product of an arbitrarily many probability spaces (§254). In the §252 the author gives Fubini’s and Tonneli’s theorems with some corollaries. Other parts of this chapter are connected with topics: convolution of functions and measures; Radon measures (absolutely continuous Radon measures).

The starting point of Chapter 26 (entitled “Change of variable in the integral”) is Vitali’s theorem in \(\mathbb{R}^{r}\), which allows to formulate the integral form for the density theorem (in \(\mathbb{R}^{r}\) (the basic information connected with density topology one can find in section 261Y: “Further exercises”). Next two paragraphs are devoted to the discussion of the following topics: Lipschitz and differentiable functions (with Rademacher’s theorem) and differentiable functions in \(\mathbb{R}^{r}\). In the §264 the author gives the definition and fundamental properties of Hausdorff \(r\)-dimensional measure in Euclidean space.

The initial paragraph of Chapter 27 entitled “Probability theory” is connected with probability distributions, while the second paragraph is devoted to (stochastically) independence of events, \(\sigma\)-algebras and random variables. The most important results of this chapter are contained in paragraphs entitled: “The strong law of large numbers” and “The central limit theorem” (this section contains Lindeberg’s theorem). This chapter completes considerations concerning martingales.

Chapter 28 (entitled “Fourier analysis”) has been divided into three parts: Fourier series, Fourier transforms and the characteristic functions of the Radon probability measure. It should be mentioned here that all these considerations have been preceded by the paragraph connected with the Stone-Weierstrass theorem (on the approximation of continuous functions) in a variety of forms. 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 (ordered sets), general topology and topology of the Euclidean space, normed spaces, linear topological spaces and factorization of matrices. Each section of this treatise ends with “basic exercises”, “further exercises” and “Notes and comments”.

First section of Chapter 21 contains a list of definitions distinguishing different types of measure spaces. In the next sections the author gives a brief account of the theory of some measure spaces described in the first section. Part 214 is connected with subspaces, which lead to the ”integration over a subset”. In the last section of this chapter the author presents some examples which allow to establish relationships between various kinds of measure spaces. The starting point of the second chapter (marked number 22) is Vitali’s theorem in \(\mathbb{R}\). The main aim of the second section of this chapter is Theorem 222E connected with differentiation of indefinite integral. Next the author proceeds to the Lebesgue’s density theorems: first in the integral form (also for complex-valued functions) and in exercises in form leading to the density topology. The last three parts of this chapter contain the most important statements concerning the functions of bounded variation and absolutely continuous functions (as an example of applications of the results: Theorem of integration by parts, theorems for lower semi-continmuous functions and the Lebesgue decomposition of a function of bounded variation).

Chapter 23 is devoted to the Radon-Nikodým theorem. The first section of this chapter introduces the basic tools: countable additive functional and its decompositions. The main theorem is contained in the second paragraph (marked number 232), which is finished with considerations concerning Lebesgue decomposition of a countably additive functional. In the next part the author gives some application of the Radon-Nikodým theorem providing the concept of “conditional expectations”. The successive sections are devoted to the following problems: indefinite-integral measures from a non-negative measurable function on a measure space; general method of integration by substitution; inverse measure preserving functions.

In Chapter 24 the author discusses function spaces \(L^{0}\), \(L^{p}\) and \(L^{\infty}\). In this field the following problems are analyzed more minutely: Riesz spaces, the space \(L^{1}\) (the norm of \(L^{1}\)), a dense subspaces of \(L^{\infty}\); norms of \(L^{p}\); convergence in measure; pointwise convergence; embedding \(L^{p}\) in \(L^{0}\); uniform integrability; compactness in \(L^{1}\). In paragraph 246Y (“Further exercise”) the Viltali-Hahn-Saks theorem is contained).

In Chapter 25 the author considered product measure: two (finite) product (§251) and the product of an arbitrarily many probability spaces (§254). In the §252 the author gives Fubini’s and Tonneli’s theorems with some corollaries. Other parts of this chapter are connected with topics: convolution of functions and measures; Radon measures (absolutely continuous Radon measures).

The starting point of Chapter 26 (entitled “Change of variable in the integral”) is Vitali’s theorem in \(\mathbb{R}^{r}\), which allows to formulate the integral form for the density theorem (in \(\mathbb{R}^{r}\) (the basic information connected with density topology one can find in section 261Y: “Further exercises”). Next two paragraphs are devoted to the discussion of the following topics: Lipschitz and differentiable functions (with Rademacher’s theorem) and differentiable functions in \(\mathbb{R}^{r}\). In the §264 the author gives the definition and fundamental properties of Hausdorff \(r\)-dimensional measure in Euclidean space.

The initial paragraph of Chapter 27 entitled “Probability theory” is connected with probability distributions, while the second paragraph is devoted to (stochastically) independence of events, \(\sigma\)-algebras and random variables. The most important results of this chapter are contained in paragraphs entitled: “The strong law of large numbers” and “The central limit theorem” (this section contains Lindeberg’s theorem). This chapter completes considerations concerning martingales.

Chapter 28 (entitled “Fourier analysis”) has been divided into three parts: Fourier series, Fourier transforms and the characteristic functions of the Radon probability measure. It should be mentioned here that all these considerations have been preceded by the paragraph connected with the Stone-Weierstrass theorem (on the approximation of continuous functions) in a variety of forms. 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 (ordered sets), general topology and topology of the Euclidean space, normed spaces, linear topological spaces and factorization of matrices. 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 |

46-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis |