Sets, Functions, Measures

480 Pages·2018·3.21 MB·English

by  Valeriy K. Zakharov, Timofey V. Rodionov, Alexander V. Mikhalev

ValeriyK.Zakharov,TimofeyV.Rodionov,AlexanderV.Mikhalev
Sets,Functions,Measures
De Gruyter Studies
in Mathematics
|
Editedby
CarstenCarstensen,Berlin,Germany
GavrilFarkas,Berlin,Germany
NicolaFusco,Napoli,Italy
FritzGesztesy,Waco,Texas,USA
NielsJacob,Swansea,UnitedKingdom
ZenghuLi,Beijing,China
Karl-HermannNeeb,Erlangen,Germany
Volume 68/2
Valeriy K. Zakharov, Timofey V. Rodionov,
Alexander V. Mikhalev
Sets, Functions,
Measures
|
Volume II: Fundamentals of Functions
and Measure Theory
MathematicsSubjectClassification2010
26-02,28-02,26A21,26A30,26A42,28A05,28A25,28C05,28C15,46E25,46J10,54A05,54C30
Author
Prof.Dr.ValeriyK.Zakharov
LomonosovMoscowStateUniversity
FacultyofMathematicsandMechanics
DepartmentofMathematicalAnalysis
LeninskieGoryb.1,GSP-1
119991Moscow
Russia
Coauthors
Prof.Dr.TimofeyV.Rodionov Prof.Dr.AlexanderV.Mikhalev
LomonosovMoscowStateUniversity LomonosovMoscowStateUniversity
FacultyofMathematicsandMechanics FacultyofMathematics&Mechanics
DepartmentofMathematicalAnalysis DepartmentofTheoreticalInformatics
LeniniskieGoryb.1,GSP-1 LeninskieGoryb.1,GSP-1
119991Moscow 119991Moscow
Russia Russia
ISBN978-3-11-055009-2
e-ISBN(PDF)978-3-11-055096-2
e-ISBN(EPUB)978-3-11-055022-1
Set-ISBN978-3-11-055097-9
ISSN0179-0986
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ACIPcatalogrecordforthisbookhasbeenappliedforattheLibraryofCongress.
BibliographicinformationpublishedbytheDeutscheNationalbibliothek
TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de.
©2018WalterdeGruyterGmbH,Berlin/Boston
Typesetting:CompuscriptLtd.,Shannon,Ireland
Printingandbinding:CPIbooksGmbH,Leck
♾Printedonacid-freepaper
PrintedinGermany
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|
The authors dedicate their book to the centenary of Felix Hausdorff’s outstanding
book“SetTheory”

Contents
HistoricalforewordonthecentenaryafterFelixHausdorff’sclassicSetTheory|xi
Preface|xv
2 Fundamentalsofthetheoryoffunctions|1
Introduction|1
2.1 Descriptiveandprescriptivespaces|2
2.1.1 Ensemblesandtheirenvelopes|2
2.1.2 Thefourtransfinitecollectionsofextensionsofanensemble|20
2.1.3 ClassificationofBorelsetsforarbitraryandperfect
ensembles|31
2.1.4 Descriptivespaceswithnegligence|42
2.1.5 Prescriptivespaces|47
2.2 Familiesofreal-valuedfunctionsonaset|49
2.2.1 Real-valuedfunctionsandpointwiseoperationsoverthem|49
2.2.2 Thepointwiseorderbetweenfunctions|52
2.2.3 Thepointwiseanduniformconvergencesofnetsandsequencesof
functions|55
2.2.4 Someusefulfunctionalfamilies|59
2.2.5 Zero-setsandcozero-setsoffunctions|71
2.2.6 Theequivalenceoffunctionswithrespecttoidealensembles|74
2.2.7 Seminormsandnormsoftheuniformconvergenceonfamilies
andfactor-familiesoffunctions|77
2.2.8 Pointwisecontinuouslinearfunctionalsonlattice-orderedlinear
spacesoffunctions|92
2.2.9 Truncatablelattice-orderedlinearspacesoffunctions|98
2.3 Familiesofmeasurableanddistributablefunctionsonadescriptive
space|100
2.3.1 Measurableanddistributablefunctions|100
2.3.2 Pointwiseoperationsovermeasurableanddistributable
functions|104
2.3.3 Thepointwiseorderbetweenmeasurableanddistributable
functions|108
2.3.4 Thepointwiseanduniformconvergencesofsequencesof
measurableanddistributablefunctions|109
2.3.5 Separabilityofsetsbymeasurableanddistributable
functions|116
viii | Contents
2.3.6 Descriptionofnormalandcompletelynormalfamiliesand
envelopes.Naturalnessofthefamilyofmeasurable
functions|119
2.3.7 CorrelationsbetweenBaire’sandBorel’sfunctional
collections|124
2.3.8 Familiesofsemimeasurablefunctionsonaspacewithan
ensemble|134
2.4 Familiesofuniformfunctionsonaprescriptivespace|140
2.4.1 Uniformfunctionsandtheirproperties|140
2.4.2 Pointwiseoperationsoveruniformfunctions|142
2.4.3 Theuniformconvergenceofsequencesofuniformfunctions|144
2.4.4 Separabilityofsetsbyuniformfunctions|146
2.4.5 Symmetrizablefunctionsonaspacewithanensemble|150
2.4.6 Descriptionsofboundedlynormalfamiliesandenvelopes.
Naturalnessofthefamilyofuniformfunctions|153
2.4.7 FinecorrelationsbetweenBaire’sandBorel’sfunctional
collections|158
2.5 Familiesoffunctionsonadescriptivespacewithnegligence|162
2.5.1 Almostmeasurable,almostdistributable,andalmostuniform
functions|162
2.5.2 Quasimeasurable,quasidistributable,andquasiuniform
functions|165
3 Fundamentalsofthemeasuretheory|179
Introduction|179
3.1 Spaceswithsemimeasuresandmeasures|180
3.1.1 Spaceswithevaluations,semimeasuresandmeasures|180
3.1.2 Familiesofevaluations,semimeasures,andmeasuresona
descriptivespace|185
3.1.3 Thetotalvariationofanaturalevaluation|187
3.1.4 Someextensionsofadditiveevaluationsdefinedonsemirings
andrings|193
3.1.5 Extensionofapositivemeasuretoawidecompletesaturated
measure|208
3.1.6 PropertiesoftheextendedBorel–LebesguemeasureonRn|222
3.2 Decompositionsofsemimeasuresandmeasures|226
3.2.1 TheHahnandJordandecompositionsofmeasures
ona𝛿-ring|226
3.2.2 TheRieszdecompositionofoverfinitesemimeasuresandmeasures
onaring|231
3.2.3 Normsonlinearspacesofboundedsemimeasuresand
measures|239
Contents | ix
3.2.4 AbsolutecontinuityandsingularityandtheLebesgue
decomposition|240
3.3 TheLebesgueintegral|247
3.3.1 Measurablefunctionsonaspacewithapositivewide
measure|247
3.3.2 TheLebesgueintegraloveraspacewithapositivemeasure|255
3.3.3 SequentialpropertiesoftheLebesgueintegral|263
3.3.4 Propertiesofdensityandcompletenessforthefamilyandthe
factor-familyofintegrablefunctions|266
3.3.5 ComparisonofsomeLebesgueintegralsoverspaceswithpositive
widemeasures|273
3.3.6 TheLebesgueintegraloveraspacewithanarbitrarywidemeasure.
TheproblemofcharacterizationofLebesgueintegralsaslinear
functionals|276
3.3.7 Widemeasuresdefinedbydensities|283
3.3.8 TheLebesgue–Radon–Nikodymtheorem|289
3.3.9 Dualtothefactor-spaceofintegrablefunctions|296
3.4 RepresentationofafunctionalbytheLebesgueintegral|300
3.4.1 Regularityandcontinuityofevaluations.Thekeytheoremfor
integralrepresentations|300
3.4.2 Representationofpointwise𝜎-continuousfunctionalsbyLebesgue
integrals.Thesolutionoftheproblemofcharacterizationof
Lebesgueintegralsaslinearfunctionals|303
3.4.3 RepresentationofpointwisecontinuousfunctionalsbyLebesgue
integrals|311
3.5 Topologicalspaceswithmeasures.TheRadonintegral|316
3.5.1 Topologicalspaceswithevaluations,semimeasures,and
measures|316
3.5.2 Measurableandintegrablefunctionsontopologicalspaceswith
measures|323
3.5.3 WideRadonmeasuresonHausdorffspaces.Theproblemof
characterizationofRadonintegralsaslinearfunctionals|328
3.5.4 NarrowRadonmeasuresonHausdorffspaces|334
3.5.5 RadonbimeasuresonHausdorffspaces|339
3.5.6 TheRadonintegraloveraHausdorffspacewithaRadon
bimeasure|352
3.6 RepresentationofafunctionalbytheRadonintegral|355
3.6.1 𝜎-Exactlinearfunctionalsonspacesofsymmetrizable
functions|355
3.6.2 Extensionsof𝜎-exactfunctionalsonspacesofsymmetrizable
functionsbytheYoung–Daniellmethod|362