The Bartle-Dunford-Schwartz integral: integration with respect to a sigma-additive vector measure

310 Pages·2008·1.94 MB·English

by  Thiruvaiyaru V. Panchapagesan

M o n o g r a f i e M a t e m a t y c z n e
Instytut Matematyczny Polskiej Akademii Nauk (IMPAN)
Volume 69
(New Series)
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S. Banach, B. Knaster, K. Kuratowski,
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Tadeusz Iwaniec (Syracuse University, USA)
Tom Körner (Cambridge, UK)
Krystyna Kuperberg (Auburn University, USA)
Tomasz Łuczak (Poznán University, Poland)
Ludomir Newelski (Wrocław University, Poland)
Gilles Pisier (Université Paris 6, France)
Piotr Pragacz (Institute of Mathematics, Polish Academy of Sciences)
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Jerzy Zabczyk (Institute of Mathematics, Polish Academy of Sciences)
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T.V. Panchapagesan
The
Bartle-Dunford-Schwartz
Integral
Integration with Respect to a Sigma-Additive Vector Measure
Birkhäuser
Basel · Boston · Berlin
Author:
T.V. Panchapagesan Formerly:
301, Shravanthi Palace Departamento de Matemáticas
3rd Cross, C.R.Layout Facultad de Ciencias
J.P. Nagar, 1st Phase Universidad de los Andes
Bangalore 560 078 Mérida 5101
India Venezuela
e-mail: [email protected] e-mail: [email protected]
2000 Mathematical Subject Classification: primary 28B05; secondary 46E40, 46G10
Library of Congress Control Number: 2007942613
Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists
this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in
the Internet at http://dnb.ddb.de
ISBN 978-3-7643-8601-6 Birkhäuser Verlag AG, Basel - Boston - Berlin
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ISBN 978-3-7643-8601-6 e-ISBN 978-3-7643-8602-3
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The monograph is dedicated to the memory
of the author’s spiritual guru
Singaracharya Swamigal
of Kondapalayam, Scholingur,
and to the memory of his eldest brother
T.V. Gopalaiyer,
a great Tamil scholar, of Ecole Fran¸caise
D’extrˆeme-Orient, Centre de Pondicherry.
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
1 Preliminaries
1.1 Banach space-valued measures . . . . . . . . . . . . . . . . . . . . 1
1.2 lcHs-valued measures . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Basic Properties of the Bartle-Dunford-Schwartz Integral
2.1 (KL) m-integrability . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 (BDS) m-integrability . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 L -spaces, 1≤p≤∞
p
3.1 The seminorms m•(·,T) on L M(m), 1≤p<∞ . . . . . . . . . 33
p p
3.2 Completeness of L M(m) and L I(m),1≤p<∞,
p p
and of L∞(m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 Characterizations of L I(m), 1≤p<∞ . . . . . . . . . . . . . . 47
p
3.4 Other convergence theorems for L (m), 1≤p<∞ . . . . . . . . . 54
p
3.5 Relations between the spaces L (m) . . . . . . . . . . . . . . . . . 63
p
4 Integration With Respect to lcHs-valued Measures
4.1 (KL) m-integrability (m lcHs-valued). . . . . . . . . . . . . . . . . 65
4.2 (BDS) m-integrability (m lcHs-valued) . . . . . . . . . . . . . . . . 78
4.3 The locally convex spaces L M(m), L M(σ(P),m),
p p
L I(m)and L I(σ(P),m), 1≤p<∞ . . . . . . . . . . . . . . . . 85
p p
4.4 Completeness of L M(m),L I(m), L M(σ(P),m)
p p p
and L I(σ(P),m), for suitable X . . . . . . . . . . . . . . . . . . . 91
p
4.5 Characterizations of L -spaces, convergence
p
theorems and relations between L -spaces . . . . . . . . . . . . . . 101
p
4.6 Separability of L (m) and L (σ(P),m),1≤p<∞,
p p
m lcHs-valued . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
viii Contents
5 Applications to Integration in Locally Compact Hausdorff Spaces – Part I
5.1 Generalizations of the Vitali-Carath´eodory
Integrability Criterion Theorem . . . . . . . . . . . . . . . . . . . . 117
5.2 The Baire version of the Dieudonn´e-Grothendieck
theorem and its vector-valued generalizations . . . . . . . . . . . . 121
5.3 Weakly compact bounded Radon operators
and prolongable Radon operators . . . . . . . . . . . . . . . . . . . 138
6 ApplicationstoIntegrationinLocallyCompactHausdorffSpaces–PartII
6.1 Generalized Lusin’s Theorem and its variants . . . . . . . . . . . . 153
6.2 Lusin measurability of functions and sets . . . . . . . . . . . . . . 159
6.3 Theorems of integrability criteria . . . . . . . . . . . . . . . . . . . 165
6.4 Additional convergence theorems . . . . . . . . . . . . . . . . . . . 187
6.5 Duals of L (m) and L (n) . . . . . . . . . . . . . . . . . . . . . . . 202
1 1
7 Complements to the Thomas Theory
7.1 Integration of complex functions with respect
to a Radon operator . . . . . . . . . . . . . . . . . . . . . . . . . . 213
7.2 Integration with respect to a weakly compact
bounded Radon operator. . . . . . . . . . . . . . . . . . . . . . . . 221
7.3 Integration with respect to a prolongable Radon operator . . . . . 233
7.4 Baire versions of Proposition 4.8 and Theorem 4.9 of [T] . . . . . . 241
7.5 Weakly compact and prolongable Radon vector measures . . . . . 251
7.6 Relation between L (u) and L (m ), u a weakly compact
p p u
bounded Radon operator or a prolongable Radon operator . . . . . 275
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
Preface
In 1953, Grothendieck [G] characterized locally convex Hausdorff spaces which
have the Dunford-Pettis property and used this property to characterize weakly
compact operators u :C(K) →F, where K is a compact Hausdorff space and F
isa locallyconvexHausdorffspace(briefly,lcHs) whichis complete.Amongother
results,healsoshowedthatthereisabijectivecorrespondencebetweenthefamily
of all F-valued weakly compact operators u on C(K) and that of all F-valued
σ-additiveBairemeasuresonK.Buthe didnotdevelopanytheoryofintegration
to represent these operators.
Later, in 1955, Bartle, Dunford, and Schwartz [BDS] developed a theory of
integration for scalar functions with respect to a σ-additive Banach-space-valued
vector measure m defined on a σ-algebra of sets and used it to give an integral
representationforweaklycompactoperatorsu:C(S)→X,whereS isacompact
Hausdorffspace andX is a Banachspace.A modified formof this theory is given
inSection10ofChapterIVof[DS1].Inhonoroftheseauthors,wecalltheintegral
introduced by them as well as its variants given in Section 2.2 of Chapter 2 and
in Section 4.2 of Chapter 4, the Bartle-Dunford-Schwartz integral or briefly, the
BDS-integral.
Aboutfifteenyearslater,in[L1,L2]LewisstudiedaPettistypeweakintegral
ofscalarfunctions with respect to a σ-additive vector measurem havingrange in
anlcHsX.ThistypeofdefinitionhasalsobeenconsideredbyKluva´nekin[K2].In
honorofthesemathematicianswecalltheintegralintroducedin[L1,L2]aswellas
its variants givenin Section 2.1 of Chapter 2 and in Section 4.1 of Chapter 4, the
Kluva´nek-Lewis integralor briefly, the (KL)-integral.When the domain of the σ-
additivevectormeasuremisaσ-algebraΣandX isaBanachspace,Theorem2.4
of [L1] asserts that the (KL)-integral is the same as the (BDS)-integral. Though
this result is true, its proof in [L1] lacks essential details. See Remark 2.2.6 of
Chapter 2.
LetT bea locallycompactHausdorffspaceandK(T)(resp.K(T,R))bethe
vector space of all complex- (resp. real-) valued continuous functions on T with
compact support, endowed with the inductive limit locally convex topology as in
§1, Chapter III of [B]. In 1970, using the results of [G], Thomas [T] developed a
theory of vectorial Radon integration with respect to a weakly compact bounded
x Preface
(resp. a prolongable) Radon operator u on K(T,R) with values in a real Banach
space and more generally, with values in a real quasicomplete lcHs. ( Thomas [T]
calls them bounded weakly compact (resp. prolongable)Radon vector measures.)
However,bymakingsomemodifications,itcanbeshownthathistheoryisequally
valid for such operators u defined on K(T) with range contained in a complex
Banach space or in a quasicomplete complex lcHs. See Section 7.1 of Chapter 7.
Functions f integrable with respect t(cid:2)o u are called u-integrable and the integral
of f with respect to u is denoted by fdu in [T].
T
If S is a compact Hausdorff space, X a Banach space and u : C(S) → X is
a weakly compact operator, then by the Bartle-Dunford-Schwartz representation
theorem in [BDS] and in [DU] there exists a unique X-valued Borel regular σ-
additive ve(cid:2)ctor measure mu on the σ-algebra B(S) of the Borel sets in S such
that uf = fdm for f ∈C(S). If X is a real Banach space and if u:Cr(S)→
S u
X (where Cr(S) = {f ∈ C(S) : freal-valued}) is a weakly compact operator,
Thomas showed in [T] that a real function f on S is u-integrable if and only if
it is m(cid:2)u-integrable in t(cid:2)he sense of Section 10 of Chapter IV of [DS1] and in that
case, fdu = (BDS) fdm . Moreover, such f is m -measurable in the sense
S S u u
ofSection10ofChapterIVof[DS1]thoughitisnotnecessarilyB(S)-measurable.
On the other hand, in [P3, P4] we have shown that there is a bijective cor-
respondence Γ between the dual space K(T)∗ and the family of all δ(C)-regular
complex measures on δ(C) such that Γθ = μθ|δ(C) for θ ∈ K(T)∗, where μθ is the
complex Radon measure determined by θ (in the sense of [P4]) and δ(C) is the
δ-ringgeneratedbythefamilyC ofallcompactsetsinT.Asobservedin[P13],the
scalar-valued prolongable Radon operators on K(T) are precisely the continuous
linear functionals on K(T) (i.e., the elements of K(T)∗).
From the above results of Thomas [T] and of Panchapagesan [P3, P4, and
P13] the following questions arise:
(Q1) SimilartoSection10ofChapterIVof[DS1],canatheoryofintegrationof
scalar functions be developed with respect to a σ-additive quasicomplete
lcHs-valued vector measure m defined on a σ-algebra or on a σ-ring S
of sets, permitting the integration of scalar functions (with respect to m)
which are not necessarily S-measurable?
(Q2) The same as in (Q1), excepting that the domainof m is a δ-ring P of sets
and the S-measurability of functions is replaced by σ(P)-measurability
(where σ(P) denotes the σ-ring generated by P).
(Q3) The Bartle-Dunford-Schwartz representation theorem has been general-
ized in [P9] for weakly compact operators on C (T) and hence for weakly
0
compact bounded Radon operatorsu on K(T) with values in a quasicom-
plete lcHs X assertingthat u determines a unique B(T)-regularX-valued
σ-additive vector measure m on B(T), the σ-algebra of Borel sets in T.
u
The question is: Is it possible to give a similar representationtheorem for
X-valued prolongable Radon operators u on K(T)?
Preface xi
(Q4) If (Q3) has an affirmative answer, does the prolongable operator u deter-
mine a σ-additive δ(C)-regular (suitably defined) vector measure m on
u
δ(C)? (This question is suggested by the scalar analogue in [P13].)
(Q5) Suppose (Q1) has an affirmative answer and suppose u:K(T)→X, X a
quasicomplete lcHs, is a weakly compact bounded Radon operator deter-
mining the σ-additive vector measure m on B(T). Can it be shown that
u
ascalarfunctionf is u-integrableinT ifandonly ifit(cid:2)ismu-int(cid:2)egrablein
T (inthesenseofintegrationgivenfor(Q1))?Ifso,is fdu= fdm ?
T T u
(Q6) If (Q2), (Q3) and (Q4) are answered in the affirmative and if u:K(T)→
X, X a quasicomplete lcHs, is a prolongableRadon operator determining
the σ-additive vector measure m on δ(C), then can it be shown that a
u
scalar function f with compact support is u-integrable in T if(cid:2)and only if
(cid:2)it is mu-integrable in T? If so, what is the relation between T fdu and
fdm ?
T u
In the literature, integration of scalar functions with respect to a Banach-space-
valued or a sequentially complete lcHs-valued σ-additive vector measure defined
on a σ-algebra Σ of sets has been studied for Σ-measurable scalar functions in
severalpapers such as [C4], [Del1, Del2, Del4], [FNR], [FMNP], [FN1,FN2], [JO],
[K1, K2, K4], [KK], [L1], [N], [O], [OR1, OR2, OR3, OR4, OR5, OR6], [Ri1, Ri2,
Ri3,Ri4,Ri5,Ri6]and[Shu1,Shu2].Asimilarstudyhasbeendonein[BD1],[L2],
[Del3],and[MN2]foraσ-additivevectormeasuredefinedonaδ-ringP.Recently,
the vector measure integration on σ-algebras has been used to study the repre-
sentation of real Banach lattices in [C1,C2, C3, C4]. Also see [CR1, CR2, CR3],
[FMNSS1,FMNSS2],[MP],[OSV],[SP1,SP2,SP3,SP4]and[St].Someotherpapers
whichcanalsobe referredfor variousaspects ofvectormeasuresdefined onrings,
δ-rings and σ-rings are [Br], [BD1], [BD2] and [Del4]. None of the above papers
considersthe possibilityofintegratingnon-Σ-measurableornon-σ(P)-measurable
functions and hence the integrationtheory developed in the literature is not suit-
able for answering the above questions. Though the paper [BD2] treats the in-
tegration of non-σ(P)-measurable functions, its results do not answer the above
questions.
However,adaptingsomeoftheconceptsandtechniquesusedbyDobrakovin
[Do1,Do2]andbyDobrakovandPanchapagesanin[DP2](inthestudyofintegra-
tion of vector functions with respect to an operator-valued measure), the present
monographanswers all the questions raised above in the affirmative. Moreover,a
nice theory of L -spaces, 1 ≤ p < ∞, is also developed for a σ-additive Banach
p
space-valued (resp. quasicomplete or sequentially complete lcHs-valued) vector
measuremdefinedonaδ-ringP ofsetsandresultssimilartothoseknownforthe
abstractLebesgue andBochnerL -spacesareobtained.Compare with[FMNSS2]
p
and [SP1].
Themonographconsistsofsevenchapters.Chapter1isonPreliminariesand
hastwosections.Section1.1isdevotedtofixingthenotationandterminologyand
to give some definitions and results from the literature on Banach space-valued