Measure, integration and function spaces.

*(English)*Zbl 0814.28001
Singapore: World Scientific. x, 277 p. (1994).

This monograph is an interesting addition to the existing literature on measure and integration theory. As indicated in the title, the book covers not only the classical topics of abstract measure and integration theory but also includes a discussion of spaces of measurable functions. The scope of the book is in fact even broader than the title suggests since about one third of the book is devoted to general normed and ordered vector spaces and to spaces of measurable functions and spaces of additive set functions.

Chapter 1 (Introduction) reviews Lebesgue’s thesis and shows very clearly how the idea of an integral as being a particular sequentially order continuous positive linear functional on a suitable (ordered) vector space of real-valued functions leads to the problem of defining a measure on a suitable class of subsets of the real line. These facts are, of course, well-known, but it is nice to have the details since the modern presentation in textbooks, proceeding from measures to integrals, is exactly opposite to the historical development of the subject. With regard of the functional analytic flavour of major parts of the book, one might, in addition, desire some comments on how the properties of spaces of measurable functions stimulated the development of functional analysis; such comments, however, are not given in this book.

Chapter 2 (Measure theory) contains the essential part of the classical material on classes of sets, set functions, existence and uniqueness of measure extensions, and the Lebesgue measure. In addition, special attention is given to finitely additive set functions, their Jordan and Yosida-Hewitt decompositions, and regularity. Without need, uniqueness of the Jordan decomposition is only established for the countably additive case and via the Hahn decomposition; also, in spite of its complete analogy with the Yosida-Hewitt decomposition, the Lebesgue decomposition is deferred to the chapter on integration, and only the countably additive case is considered there. The chapter ends with the Nikodým convergence and boundedness theorems for signed measures on a \(\sigma\)- algebra.

Chapter 3 (Integration) concerns measurable functions, the Lebesgue integral, and the Radon-Nikodým theorem (for a \(\sigma\)-finite measure). One of the first results is the monotone approximation of measurable functions by simple functions, which is the key to the subsequent construction of the Lebesgue integral. The comparison of the Lebesgue and Riemann integrals is rather short, but there are some interesting complements like Mikusiński’s characterization of a Lebesgue integrable function, which is used to prove Fubini’s theorem, and the geometric interpretation of the indefinite Lebesgue integral of a positive function as a product measure acting on the product of the domain and the range of the function. The chapter ends with the Vitali- Hahn-Saks theorem for signed measures on a \(\sigma\)-algebra (which is in no way related to integration theory).

Chapter 4 (Differentiation and integration) develops differentiation theory for the Lebesgue integral, differentiation of monotone functions, and the fundamental theorem of calculus for the Lebesgue integral. The final section focusses on absolutely continuous functions.

Chapter 5 (Introduction to functional analysis) starts with normed vector spaces, linear mappings, and quotient spaces; it also presents the uniform boundedness principle, the closed graph and open mapping theorems, and the Hahn-Banach theorem with several applications, including the extension of a bounded additive set function from an algebra to the generated \(\sigma\)-algebra. The final section gives an introduction to ordered vector spaces, in particular Riesz spaces and Banach lattices; however, the famous and rather elementary band decomposition of a Riesz space and the notion of a sequentially order continuous functional are not considered at all, although these concepts are utmost important with regard to decompositions of set functions, and the monotone convergence theorem, respectively. Most examples in this chapter concern sequence spaces, which are only of marginal interest in integration theory.

Chapter 6 (Function spaces) deals primarily with \(L^ p\)-spaces \((1\leq p\leq \infty)\) and their duality theory (for a \(\sigma\)-finite measure in the case \(p= 1\)) and with spaces of bounded additive set functions and of finite signed measures. It also contains the Riesz representation theorem for positive linear functionals on the space of all continuous functions on a locally compact Hausdorff space which have compact support. The final section presents basic facts on Hilbert spaces, including the projection theorem for a closed convex subset and the Fourier expansion; moreover, it includes a discussion of the Fourier transform of functions on the real line.

The Appendix collects several results on functions of bounded variation, the Baire category theorem, the Arzelà-Ascoli theorem, and the Stone- Weierstrass theorem. Numerous exercises are provided throughout the book.

The present monograph benefits from its emphasis on additive set functions and functional analytic aspects, but at the same time it suffers from an unfortunate organisation of part of the material. In the reviewer’s opinion, purely measure theoretic or functional analytic considerations should not penetrate the discussion of integration theory, and general functional analysis should be clearly separated from the discussion of special spaces occurring in measure and integration theory.

The book also contains several inaccuracies. A mysterious statement, accompanied by an example, asserts that it is not, in general, true that the composition of measurable functions is measurable (p. 73). Also, the Lebesgue integral is first defined with respect to an arbitrary measure, but later this notion is restricted to the integral with respect to Lebesgue measure (pp. 81 and 85). Moreover, there is no clear distinction between the semi-normed vector spaces of integrable functions and the normed vector spaces of their equivalence classes; this affects the definitions and the use of the notions of the dual and of reflexivity (Chapters 5 and 6).

The book suffers from various deficiencies also from a purely technical point of view: The form of references to formulas and results is inconsistent (pp. 3 and 29), Theorem 14 of Section 5.1 is stated twice (pp. 169 and 170), abbreviations of items in the bibliography (which is rather selective) do not match the alphabetical order, the list of symbols employs an absolutely incomprehensible ordering which is neither alphanumeric nor by occurrence, and the combined author and subject index fails to be alphabetically ordered and contains several incorrect page numbers.

In spite of its deficiencies which are regrettable, this monograph can be recommended to be read with caution but as a source of inspiration.

Chapter 1 (Introduction) reviews Lebesgue’s thesis and shows very clearly how the idea of an integral as being a particular sequentially order continuous positive linear functional on a suitable (ordered) vector space of real-valued functions leads to the problem of defining a measure on a suitable class of subsets of the real line. These facts are, of course, well-known, but it is nice to have the details since the modern presentation in textbooks, proceeding from measures to integrals, is exactly opposite to the historical development of the subject. With regard of the functional analytic flavour of major parts of the book, one might, in addition, desire some comments on how the properties of spaces of measurable functions stimulated the development of functional analysis; such comments, however, are not given in this book.

Chapter 2 (Measure theory) contains the essential part of the classical material on classes of sets, set functions, existence and uniqueness of measure extensions, and the Lebesgue measure. In addition, special attention is given to finitely additive set functions, their Jordan and Yosida-Hewitt decompositions, and regularity. Without need, uniqueness of the Jordan decomposition is only established for the countably additive case and via the Hahn decomposition; also, in spite of its complete analogy with the Yosida-Hewitt decomposition, the Lebesgue decomposition is deferred to the chapter on integration, and only the countably additive case is considered there. The chapter ends with the Nikodým convergence and boundedness theorems for signed measures on a \(\sigma\)- algebra.

Chapter 3 (Integration) concerns measurable functions, the Lebesgue integral, and the Radon-Nikodým theorem (for a \(\sigma\)-finite measure). One of the first results is the monotone approximation of measurable functions by simple functions, which is the key to the subsequent construction of the Lebesgue integral. The comparison of the Lebesgue and Riemann integrals is rather short, but there are some interesting complements like Mikusiński’s characterization of a Lebesgue integrable function, which is used to prove Fubini’s theorem, and the geometric interpretation of the indefinite Lebesgue integral of a positive function as a product measure acting on the product of the domain and the range of the function. The chapter ends with the Vitali- Hahn-Saks theorem for signed measures on a \(\sigma\)-algebra (which is in no way related to integration theory).

Chapter 4 (Differentiation and integration) develops differentiation theory for the Lebesgue integral, differentiation of monotone functions, and the fundamental theorem of calculus for the Lebesgue integral. The final section focusses on absolutely continuous functions.

Chapter 5 (Introduction to functional analysis) starts with normed vector spaces, linear mappings, and quotient spaces; it also presents the uniform boundedness principle, the closed graph and open mapping theorems, and the Hahn-Banach theorem with several applications, including the extension of a bounded additive set function from an algebra to the generated \(\sigma\)-algebra. The final section gives an introduction to ordered vector spaces, in particular Riesz spaces and Banach lattices; however, the famous and rather elementary band decomposition of a Riesz space and the notion of a sequentially order continuous functional are not considered at all, although these concepts are utmost important with regard to decompositions of set functions, and the monotone convergence theorem, respectively. Most examples in this chapter concern sequence spaces, which are only of marginal interest in integration theory.

Chapter 6 (Function spaces) deals primarily with \(L^ p\)-spaces \((1\leq p\leq \infty)\) and their duality theory (for a \(\sigma\)-finite measure in the case \(p= 1\)) and with spaces of bounded additive set functions and of finite signed measures. It also contains the Riesz representation theorem for positive linear functionals on the space of all continuous functions on a locally compact Hausdorff space which have compact support. The final section presents basic facts on Hilbert spaces, including the projection theorem for a closed convex subset and the Fourier expansion; moreover, it includes a discussion of the Fourier transform of functions on the real line.

The Appendix collects several results on functions of bounded variation, the Baire category theorem, the Arzelà-Ascoli theorem, and the Stone- Weierstrass theorem. Numerous exercises are provided throughout the book.

The present monograph benefits from its emphasis on additive set functions and functional analytic aspects, but at the same time it suffers from an unfortunate organisation of part of the material. In the reviewer’s opinion, purely measure theoretic or functional analytic considerations should not penetrate the discussion of integration theory, and general functional analysis should be clearly separated from the discussion of special spaces occurring in measure and integration theory.

The book also contains several inaccuracies. A mysterious statement, accompanied by an example, asserts that it is not, in general, true that the composition of measurable functions is measurable (p. 73). Also, the Lebesgue integral is first defined with respect to an arbitrary measure, but later this notion is restricted to the integral with respect to Lebesgue measure (pp. 81 and 85). Moreover, there is no clear distinction between the semi-normed vector spaces of integrable functions and the normed vector spaces of their equivalence classes; this affects the definitions and the use of the notions of the dual and of reflexivity (Chapters 5 and 6).

The book suffers from various deficiencies also from a purely technical point of view: The form of references to formulas and results is inconsistent (pp. 3 and 29), Theorem 14 of Section 5.1 is stated twice (pp. 169 and 170), abbreviations of items in the bibliography (which is rather selective) do not match the alphabetical order, the list of symbols employs an absolutely incomprehensible ordering which is neither alphanumeric nor by occurrence, and the combined author and subject index fails to be alphabetically ordered and contains several incorrect page numbers.

In spite of its deficiencies which are regrettable, this monograph can be recommended to be read with caution but as a source of inspiration.

Reviewer: Klaus D.Schmidt (Dresden)

##### MSC:

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

46E27 | Spaces of measures |

46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |