##
**Pettis integral and measure theory.**
*(English)*
Zbl 0582.46049

Mem. Am. Math. Soc. 307, 224 p. (1984).

Since the original B. J. Pettis paper [Trans. Am. Math. Soc. 44, 277-304 (1938; Zbl 0019.41603)] there had been no significant progress in the study of the Pettis integral until 1976. The integral was known only to be countably additive, absolutely continuous and of \(\sigma\)-finite variation [according to J. Diestel and J. J. Uhl, Lect. Notes Math. 1033, 144-192 (1983; Zbl 0521.46035), p. 148]. The last property was discovered by Rybakov only in 1971. The series of successful attacks on the Pettis integral started with the work of K. Musiał [Preprint Series No.7 Aarhus Uni. 50 p. (1977; Zbl 0345.46039)] where he proved that for a separable Banach space X the conjugate space \(X^*\) has the weak Radon Nikodym property (RNP) if and only if X does not contain any isomorphic copy of \(\ell_ 1\). The final version of this paper published in 1979 [Stud. Math. 64, 151-173 (1979; Zbl 0405.46015)] contained already the above theorem in the full generality (i.e. without the separability assumption). Soon other papers concerning the Pettis integration appeared: C. Stegall proved that the range of the Pettis integral defined on a perfect measure space was norm relatively compact [Math. Z. 168, 117-142 (1978; Zbl 0393.28005)], D. N. Fremlin and M. Talagrand [ibid.] constructed an example of a Pettis integrable function with non-separable range of its integral, G. A. Edgar [Indiana Univ. Math. J. 28, 559-579 (1979; Zbl 0418.46034)] published his investigations on the Pettis integral property. In the same year in Oberwolfach Edgar constructed an elementary (i.e. without additional axioms) example of a Banach space without the PIP and Musiał presented characterizations of the weak RNP in terms of the Lebesgue measure and in terms of martingales of Pettis integrable functions. And so on. The Pettis integral has opened its horn of plenty.

The book under review is a self-contained account of the most of what is new known about the Pettis integration and the measure theory tools needed for the detailed analysis of the subject. Many of the results that appear in the book were proved by the author himself or with the cooperation with other mathematicians.

Throughout the review \((X,\Sigma,\mu)\) is a complete finite measure space, E is a Banach space, (E,weak) denotes E endowed with the weak topology and Ba(E,weak) is the Baire \(\sigma\)-algebra in the weak topology.

The book is divided into 16 chapters. Chapter 1 gives the measure theory prerequisites. Besides the standard facts it contains basic results on perfect measures, liftings and the Stonian transform. The second chapter presents basic facts concerning measure theory in a Banach space. It includes in particular Edgar’s description of Ba(E,weak) in terms of \(X^*\) only, a proof of the universal measurability of (E,weak) and several properties of the Hewitt real compactification of (E,weak). Chapter 3 is devoted to the study of scalarly measurable functions. It contains a few examples of scalarly measurable functions possessing different pathological properties from the strong measurability point of view. The central part of this chapter is section 4 where the detailed study of scalarly measurable functions via their image measures on Ba(E,weak) is presented. All the results circulate on the question when a (weak*) scalarly measurable function is (weak*) scalarly equivalent to a strongly measurable one. In particular a new proof of Edgar’s characterization of Banach spaces with all scalarly measurable functions equivalent to strongly measurable ones in terms of the measure compactness of (E,weak) is given. Stonian transform of (weak*) scalarly measurable functions is in permanent use here.

The study of integration starts in Chapter 4. For a scalarly bounded and scalarly measurable \(\Phi\) : \(X\to E\) one defines a Dunford operator \(T: L^{\infty}(\mu)\to E^{**}\) associated to \(\Phi\) by setting \[ <T(f),x>=\int fx^*\circ \Phi d\mu, \] for all \(f\in L^{\infty}(\mu)\) and \(x^*\in E^*\). The main problem of the chapter is the question when T is compact (equivalently: when the vector measure defined by the Dunford integral of \(\Phi\) with respect to \(\mu\) has norm relatively compact range). The mentioned above example of Fremlin and Talagrand is described here. However, if one really wants to understand it then one should first read the whole Chapter 13.

Chapter 5 gives criteria for Pettis integrability which depend on the Baire image measure and the range of the Stonian transform. The main result is the following ”Core Theorem” (Theorem 5-2-2): If \(\Phi\) : \(X\to E\) is scalarly measurable and scalarly bounded then \(\Phi\) is Pettis integrable if and only if \(Core_{\Phi}(A)\neq \emptyset\) for all \(A\in \Sigma\) of positive measure \((Core_{\Phi}(A)\) is the intersection of all sets of the form \(\overline{conv} \Phi (A\setminus B)\), where B runs over all \(\mu\)-null sets).

Here are two important consequences of this theorem: 1. (Theorem 5-3-1) Assume that \((\Phi_ n)\) is a sequence of E-valued Pettis integrable functions and let \(\Phi\) : \(X\to E\) be scalarly integrable. Assume moreover that: a) For every \(x^*\in E^*\) we have \(x^*\circ \Phi =0\) a.e. whenever \(x^*\circ \Phi_ n=0\) a.e. for every n, b) The set \(\{x^*\circ \Phi_ n:\) \(n\in N\), \(\| x^*\| \leq 1\}\) is equi- integrable. Then \(\Phi\) is Pettis integrable.

2. (Theorem 5-3-2) Let \(\Phi\) : \(X\to E\) be scalarly measurable. Then \(\Phi\) is Pettis integrable and \(Z_{\Phi}=\{x^*\circ \Phi:\) \(\| x^*\| \leq 1\}\) is separable in \(L_ 1(\mu)\) if and only if there is a sequence \((\phi_ n)\) of E-valued simple functions such that \(x^*\circ \Phi_ n\to x^*\circ \Phi\) for each \(x^*\in E^*\) and that \(\{x^*\circ \Phi_ n:\) \(n\in N\), \(\| x^*\| \leq 1\}\) is equi-integrable.

In chapter 6 there are considered those functions \(\Phi\) : \(X\to E\) for which \(Z_{\Phi}\) is stable. It is shown that for such functions the equi-integrability of \(Z_{\Phi}\) is sufficient for being Pettis integrable. Moreover, the long-standing question of whether a bounded Pettis integrable function has always the conditional expectation is solved in the negative by presenting two examples. The results of this chapter are essentially straightforward applications of the measure- theoretic results proved only in chapters 8 to 13. Chapter 7 deals with the WRNP and the \(W^*RNP\) (the last property means that E-valued measures of finite variation have \(E^{**}\)-valued Pettis integrable densities). It is shown that the WRNP and \(W^*RNP\) can be defined in terms of the Lebesgue measure [this result of the reviewer is mistakenly quoted as published in Proc. Am. Math. Soc. It has been published in ”Topology and measure III”, Vitte/Hiddensee, GDR, 1980, Part 2, 187-191 (1982; Zbl 0515.46015)], and detailed analysis of the WRNP of conjugate spaces and weak* compact sets is reported. Assuming Axiom L, the author proves also that E has the \(W^*RNP\) if and only if for each E-valued measure of finite variation there exists a Banach space \(F\supset E\) and an F-valued Pettis integrable density of the measure. This answers a question of the reviewer [Lect. Notes Math. 794, 324-334 (1980; Zbl 0433.28010)]. Starting from Chapter 8 a thoughtful investigation of pointwise compact sets of measurable functions is presented. Chapter 8 contains Fremlin’s subsequence theorem and its proof based on non- measurable filter theorem of Talagrand (proved in Chapter 13). Detailed study of stable sets is given in Chapter 9.

Let now Y be a pointwise compact set of measurable functions on (X,\(\Sigma\),\(\mu)\) and \(\nu\) be a Radon probability on Y. The problem of whether the evaluation \((x,y)\to y(x)\) is \(\mu\) \(\times \nu\) measurable is studied in Chapter 10. It is shown that in general it is not true, but it is so for stable Y. Chapter 11 gives a technical characterization of stable sets. As consequences one gets the stability of uniformly bounded stable sets under creating convex hulls and conditional expectations. The characterization is also applied to the examination of the \(W^*RNP\) (in Chapter 7).

In Chapter 12 the following problem of A. Bellow is investigated: Let Z be a pointwise compact and Hausdorff in measure family of measurable functions on (X,\(\Sigma\),\(\mu)\). Does it follow that the topology of the convergence in measure and the topology of pointwise convergence coincide on Z ? It is proved that if \(\mu\) is perfect, or properly based, or Z is stable, or CH holds, or Axiom \(L_ 1\) is satisfied \((=\) the union of \(\aleph_ 1\) closed sets of Lebesgue measure zero is of Lebesgue measure zero), then the answer is affirmative. The main result of Chapter 13 is a construction of an extension of the Haar measure on \(\{0,1\}^ I\) to a \(\sigma\)-algebra containing all free filters on I. This is the example used in Chapter 4 in the construction of the Pettis integrable function with non-norm relatively compact range of its indefinite integral. Sets of continuous functions which are pointwise relatively compact in some spaces of measurable functions are investigated in Chapter 14. The following constitutes part of the main theorem (used in Chapter 7):

For a bounded \(A\subset C(K)\) the following conditions are equivalent:

a) A does not contain any \(\ell_ 1\)-sequence.

b) Each sequence in A contains a pointwise converging subsequence.

c) For each Radon probability \(\mu\) on K, A is stable with respect to \(\mu\).

In Chapter 15 Riemann-measurable functions \((=\) locally bounded with negligible sets of discontinuity points) on a locally compact group are considered. Chapter 16 contains further investigations of measurability in Banach spaces. In particular the problem of the equality of weak and norm Borel sets is considered and several consequences of the existence of real measurable cardinals are presented.

Remarks. 1. In spite of being self-contained the book needs hard work of the reader. One of the reasons is that it contains a lot of misprints sometimes essentially changing the sense of statements.

2. In theorem 5-3-1 one has to assume the scalar integrability of \(\Phi\). Otherwise the theorem is false even in the one dimensional case.

3. Theorem 5-3-1 in the form of the Vitali convergence theorem for the Pettis integral has been independently and by using completely different methods proved by the reviewer [Pettis integration, Proc. 13th Winter School on Abstract Analysis, Srni (Czechoslovakia), January 1985, in Rend. Circ. Mat. Palermo, to appear]. Also the characterization of Pettis integrable functions by step functions (Theorem 5-3-2) and the non- existence of the conditional expectation (Example 6-4-3) has been independently proved by the reviewer (ibid.).

4. There is a natural martingale characterization of the WRNP [Musiał, Lect. Notes Math. 794 (loc. cit.)]. I think, that it should have been inserted in this book, the more so as a martingale characterization of the \(W^*RNP\) is given (under Axiom L).

5. Recently Drewnowski has given a different proof of Theorem 5-2-2 (in the form presented in this review).

6. Probably the first example of a Pettis integrable function without conditional expectation was published by V. I. Rybakov [Mat. Zametki 10, 565-570 (1971; Zbl 0271.60005)]. Unfortunately, the function was unbounded.

7. By my opinion the expression ”the weak* RNP” is not the best one. This term exists already in the literature and denotes the following: \(E^*\) has the \(W^*RNP\) if each \(E^*\)-valued measure of finite variation has a Gelfand integrable density. Once I have proposed the term ”E has the WRNP in \(E^{**}\)”. It is longer but not misleading.

The book under review is a self-contained account of the most of what is new known about the Pettis integration and the measure theory tools needed for the detailed analysis of the subject. Many of the results that appear in the book were proved by the author himself or with the cooperation with other mathematicians.

Throughout the review \((X,\Sigma,\mu)\) is a complete finite measure space, E is a Banach space, (E,weak) denotes E endowed with the weak topology and Ba(E,weak) is the Baire \(\sigma\)-algebra in the weak topology.

The book is divided into 16 chapters. Chapter 1 gives the measure theory prerequisites. Besides the standard facts it contains basic results on perfect measures, liftings and the Stonian transform. The second chapter presents basic facts concerning measure theory in a Banach space. It includes in particular Edgar’s description of Ba(E,weak) in terms of \(X^*\) only, a proof of the universal measurability of (E,weak) and several properties of the Hewitt real compactification of (E,weak). Chapter 3 is devoted to the study of scalarly measurable functions. It contains a few examples of scalarly measurable functions possessing different pathological properties from the strong measurability point of view. The central part of this chapter is section 4 where the detailed study of scalarly measurable functions via their image measures on Ba(E,weak) is presented. All the results circulate on the question when a (weak*) scalarly measurable function is (weak*) scalarly equivalent to a strongly measurable one. In particular a new proof of Edgar’s characterization of Banach spaces with all scalarly measurable functions equivalent to strongly measurable ones in terms of the measure compactness of (E,weak) is given. Stonian transform of (weak*) scalarly measurable functions is in permanent use here.

The study of integration starts in Chapter 4. For a scalarly bounded and scalarly measurable \(\Phi\) : \(X\to E\) one defines a Dunford operator \(T: L^{\infty}(\mu)\to E^{**}\) associated to \(\Phi\) by setting \[ <T(f),x>=\int fx^*\circ \Phi d\mu, \] for all \(f\in L^{\infty}(\mu)\) and \(x^*\in E^*\). The main problem of the chapter is the question when T is compact (equivalently: when the vector measure defined by the Dunford integral of \(\Phi\) with respect to \(\mu\) has norm relatively compact range). The mentioned above example of Fremlin and Talagrand is described here. However, if one really wants to understand it then one should first read the whole Chapter 13.

Chapter 5 gives criteria for Pettis integrability which depend on the Baire image measure and the range of the Stonian transform. The main result is the following ”Core Theorem” (Theorem 5-2-2): If \(\Phi\) : \(X\to E\) is scalarly measurable and scalarly bounded then \(\Phi\) is Pettis integrable if and only if \(Core_{\Phi}(A)\neq \emptyset\) for all \(A\in \Sigma\) of positive measure \((Core_{\Phi}(A)\) is the intersection of all sets of the form \(\overline{conv} \Phi (A\setminus B)\), where B runs over all \(\mu\)-null sets).

Here are two important consequences of this theorem: 1. (Theorem 5-3-1) Assume that \((\Phi_ n)\) is a sequence of E-valued Pettis integrable functions and let \(\Phi\) : \(X\to E\) be scalarly integrable. Assume moreover that: a) For every \(x^*\in E^*\) we have \(x^*\circ \Phi =0\) a.e. whenever \(x^*\circ \Phi_ n=0\) a.e. for every n, b) The set \(\{x^*\circ \Phi_ n:\) \(n\in N\), \(\| x^*\| \leq 1\}\) is equi- integrable. Then \(\Phi\) is Pettis integrable.

2. (Theorem 5-3-2) Let \(\Phi\) : \(X\to E\) be scalarly measurable. Then \(\Phi\) is Pettis integrable and \(Z_{\Phi}=\{x^*\circ \Phi:\) \(\| x^*\| \leq 1\}\) is separable in \(L_ 1(\mu)\) if and only if there is a sequence \((\phi_ n)\) of E-valued simple functions such that \(x^*\circ \Phi_ n\to x^*\circ \Phi\) for each \(x^*\in E^*\) and that \(\{x^*\circ \Phi_ n:\) \(n\in N\), \(\| x^*\| \leq 1\}\) is equi-integrable.

In chapter 6 there are considered those functions \(\Phi\) : \(X\to E\) for which \(Z_{\Phi}\) is stable. It is shown that for such functions the equi-integrability of \(Z_{\Phi}\) is sufficient for being Pettis integrable. Moreover, the long-standing question of whether a bounded Pettis integrable function has always the conditional expectation is solved in the negative by presenting two examples. The results of this chapter are essentially straightforward applications of the measure- theoretic results proved only in chapters 8 to 13. Chapter 7 deals with the WRNP and the \(W^*RNP\) (the last property means that E-valued measures of finite variation have \(E^{**}\)-valued Pettis integrable densities). It is shown that the WRNP and \(W^*RNP\) can be defined in terms of the Lebesgue measure [this result of the reviewer is mistakenly quoted as published in Proc. Am. Math. Soc. It has been published in ”Topology and measure III”, Vitte/Hiddensee, GDR, 1980, Part 2, 187-191 (1982; Zbl 0515.46015)], and detailed analysis of the WRNP of conjugate spaces and weak* compact sets is reported. Assuming Axiom L, the author proves also that E has the \(W^*RNP\) if and only if for each E-valued measure of finite variation there exists a Banach space \(F\supset E\) and an F-valued Pettis integrable density of the measure. This answers a question of the reviewer [Lect. Notes Math. 794, 324-334 (1980; Zbl 0433.28010)]. Starting from Chapter 8 a thoughtful investigation of pointwise compact sets of measurable functions is presented. Chapter 8 contains Fremlin’s subsequence theorem and its proof based on non- measurable filter theorem of Talagrand (proved in Chapter 13). Detailed study of stable sets is given in Chapter 9.

Let now Y be a pointwise compact set of measurable functions on (X,\(\Sigma\),\(\mu)\) and \(\nu\) be a Radon probability on Y. The problem of whether the evaluation \((x,y)\to y(x)\) is \(\mu\) \(\times \nu\) measurable is studied in Chapter 10. It is shown that in general it is not true, but it is so for stable Y. Chapter 11 gives a technical characterization of stable sets. As consequences one gets the stability of uniformly bounded stable sets under creating convex hulls and conditional expectations. The characterization is also applied to the examination of the \(W^*RNP\) (in Chapter 7).

In Chapter 12 the following problem of A. Bellow is investigated: Let Z be a pointwise compact and Hausdorff in measure family of measurable functions on (X,\(\Sigma\),\(\mu)\). Does it follow that the topology of the convergence in measure and the topology of pointwise convergence coincide on Z ? It is proved that if \(\mu\) is perfect, or properly based, or Z is stable, or CH holds, or Axiom \(L_ 1\) is satisfied \((=\) the union of \(\aleph_ 1\) closed sets of Lebesgue measure zero is of Lebesgue measure zero), then the answer is affirmative. The main result of Chapter 13 is a construction of an extension of the Haar measure on \(\{0,1\}^ I\) to a \(\sigma\)-algebra containing all free filters on I. This is the example used in Chapter 4 in the construction of the Pettis integrable function with non-norm relatively compact range of its indefinite integral. Sets of continuous functions which are pointwise relatively compact in some spaces of measurable functions are investigated in Chapter 14. The following constitutes part of the main theorem (used in Chapter 7):

For a bounded \(A\subset C(K)\) the following conditions are equivalent:

a) A does not contain any \(\ell_ 1\)-sequence.

b) Each sequence in A contains a pointwise converging subsequence.

c) For each Radon probability \(\mu\) on K, A is stable with respect to \(\mu\).

In Chapter 15 Riemann-measurable functions \((=\) locally bounded with negligible sets of discontinuity points) on a locally compact group are considered. Chapter 16 contains further investigations of measurability in Banach spaces. In particular the problem of the equality of weak and norm Borel sets is considered and several consequences of the existence of real measurable cardinals are presented.

Remarks. 1. In spite of being self-contained the book needs hard work of the reader. One of the reasons is that it contains a lot of misprints sometimes essentially changing the sense of statements.

2. In theorem 5-3-1 one has to assume the scalar integrability of \(\Phi\). Otherwise the theorem is false even in the one dimensional case.

3. Theorem 5-3-1 in the form of the Vitali convergence theorem for the Pettis integral has been independently and by using completely different methods proved by the reviewer [Pettis integration, Proc. 13th Winter School on Abstract Analysis, Srni (Czechoslovakia), January 1985, in Rend. Circ. Mat. Palermo, to appear]. Also the characterization of Pettis integrable functions by step functions (Theorem 5-3-2) and the non- existence of the conditional expectation (Example 6-4-3) has been independently proved by the reviewer (ibid.).

4. There is a natural martingale characterization of the WRNP [Musiał, Lect. Notes Math. 794 (loc. cit.)]. I think, that it should have been inserted in this book, the more so as a martingale characterization of the \(W^*RNP\) is given (under Axiom L).

5. Recently Drewnowski has given a different proof of Theorem 5-2-2 (in the form presented in this review).

6. Probably the first example of a Pettis integrable function without conditional expectation was published by V. I. Rybakov [Mat. Zametki 10, 565-570 (1971; Zbl 0271.60005)]. Unfortunately, the function was unbounded.

7. By my opinion the expression ”the weak* RNP” is not the best one. This term exists already in the literature and denotes the following: \(E^*\) has the \(W^*RNP\) if each \(E^*\)-valued measure of finite variation has a Gelfand integrable density. Once I have proposed the term ”E has the WRNP in \(E^{**}\)”. It is longer but not misleading.

Reviewer: K.Musiał

### MSC:

46G10 | Vector-valued measures and integration |

46B22 | Radon-Nikodým, Kreĭn-Milman and related properties |

28A20 | Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence |

46G12 | Measures and integration on abstract linear spaces |

46B42 | Banach lattices |

46B10 | Duality and reflexivity in normed linear and Banach spaces |

28B05 | Vector-valued set functions, measures and integrals |

28A51 | Lifting theory |

03E50 | Continuum hypothesis and Martin’s axiom |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |