Measure and Integration: Publications 1997-2011

517 Pages·2012·16.82 MB·English

by  Heinz König (auth.)

Heinz König
Measure and Integration
Publications 1997–2011
Heinz König
Department of Mathematics
Universität des Saarlandes
Saarbrücken
Germany
ISBN 978-3-0348-0381-6 ISBN 978-3-0348-0382-3 (eBook)
DOI 10.1007/978-3-0348-0382-3
Springer Basel Heidelberg New York Dordrecht London

Library of Congress Control Number: 2012938961

Mathematics Subject Classification (2010): 28-02, 60A10, 60G05

© Springer Basel 2012
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of
the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or
information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar
methodology now known or hereafter developed. Exempted from this legal reservation are brief
excerpts in connection with reviews or scholarly analysis or material supplied specifically for the
purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the
work. Duplication of this publication or parts thereof is permitted only under the provisions of the
Copyright Law of the Publisher’s location, in its current version, and permission for use must always
be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright
Clearance Center. Violations are liable to prosecution under the respective Copyright Law.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publication does not imply, even in the absence of a specific statement, that such names are exempt
from the relevant protective laws and regulations and therefore free for general use.
While the advice and information in this book are believed to be true and accurate at the date of
publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for
any errors or omissions that may be made. The publisher makes no warranty, express or implied, with
respect to the material contained herein.
Printedonacid-freepaper
Springer Basel AG is part of Springer Science+Business Media (www.birkhauser-science.com)
Preface
The 25 articles of 1997–2011appearing in this volume have been selected to ex-
tendanddeveloptheauthor’sbookentitledMeasureandIntegration:AnAdvanced
Course in Basic Concepts and Applications, Springer 1997 with reprint 2009,
henceforthcalled MI. The entire work is far differentfrom other familiar texts in
thefield:ThebasisinMI ChaptersI-IIIconsistsofouterandinnerregularexten-
sionproceduresfromplainso-calledpremeasurestotheiruniquemaximalmeasure
extensions,inbothcasesinthreecolumns(cid:129) = (cid:2)στ,thefinite,sequential,andnon-
sequentialones.Thebasicweaponsaretheouterandinner(cid:129)envelopes,whichare
designedtotaketheplaceoftraditionalformationsliketheCarathe´odoryoutermea-
sures.
The new concepts and procedures led to essential improvements. Thus for the
first time the abstract theory contains the topological theory as an explicit spe-
cial case. The new envelopes allow a much simpler and truly explicit treatment,
andincorporationofthenonsequentialandtheinnerregularversionsleadtomuch
morecomprehensiveresults.Immediateemphaticaffirmationcamefromhighlyre-
spected authoritieslike Gustave Choquetand Paul Halmos (whose reaction to the
bookwasapicturepostcardwiththesolewordsCongratulationsandmanythanks,
Paul Halmos). However, the new concepts and procedures did not find their way
intosubsequenttextbooks.Theauthorsometimesheardcomplaintsabouthissevere
style.Butthemostplausiblereasonfortheirfailureseemstobethatthefoundations
inMIChaptersI-IIIcameinaversionwhichprovedtobefarmorecomprehensive
andhencemorecomplicatedthanneededinthesubsequentprincipalapplications–
infact,theywerefashionedforisotonereal-valuedsetfunctionsinsteadofnonneg-
ativeones,inthebeliefthatthiswouldbethescopeofthefuture.Thishasnowbeen
remediedwiththenewpaper(24).
The order of the articles is according to their dates of appearance, and hence
notrelatedtotheirsubstance.Thesubstantialconnectionswill bedescribedinthe
Introductionbelow.AttheendofthevolumethereisanIndexofbasicconceptsand
resultsforboththetraditionalandthenewtheories,andalistofErrataetAddenda
forthearticles.
v
vi Preface
Atpresentitappearsmostnaturalthattheauthorhimselfshouldwriteanewbook
instead.Butthisisnotpossibleinviewofmyageof82andaccompanyinghealth
problems.ThereforeIampleasedandgratefulthatBirkha¨user decided to unite the
25 articles in a collective volume.My particularthanksgo to Dorothy Mazlum for
herefficientandfriendlyeditorialwork.
January2012 HeinzKo¨nig
Contents
Preface .....................................................................................................................v
Introduction ............................................................................................................ix
Publications 1997– 2011
1. Image measures and the so-called image measure catastrophe .........................1
2. The product theory for inner premeasures .......................................................17
3. Measure and Integration: Mutual generation of outer and inner
pr emeasures .. ...................................................................................................33
4. Measure and Integration: Integral representations of isotone functionals ......57
5. Measure and Integration: Comparison of old and new procedures .................89
6. What are signed contents and measures? ......................................................103
7. Upper envelopes of inner premeasures ..........................................................127
8. On the inner Daniell-Stone and Riesz representation theorems ....................149
9. Sublinear functionals and conical measures ..................................................165
10. Measure and Integration: An attempt at unified systematization ..................175
11. New facts around the Choquet integral .........................................................235
12. The (sub/super)additivity assertion of Choquet ............................................245
13. Projective limits via inner premeasures and the true Wiener measure ..........273
14. Stochastic processes in terms of inner premeasures ......................................313
15. New versions of the Radon-Nikodým theorem .............................................343
16. The Lebesgue decomposition theorem for arbitrary contents .......................353
17. The new maximal measures for stochastic processes ....................................369
18. Stochastic processes on the basis of new measure theory .............................391
19. New versions of the Daniell-Stone-Riesz representation theorem ................405
vii
viii Contents
20. Measure and Integral: New foundations after one hundred years .................419
21. Fubini-Tonelli theorems on the basis of inner and outer premeasures ..........437
22. Measure and Integration: Characterization of the new maximal contents
and measures ..................................................................................................455
23. Notes on the projective limit theorem of Kolmogorov .................................465
24. Measure and Integration: The basic extension theorems ..............................479
25. Measure Theory: Transplantation theorems for inner premeasures ..............497
Index ....................................................................................................................505
Sources and Permissions.......................................................................................507
Errata et Addenda .................................................................................................E1
Introduction
Thepresentarticleshavebeenwritten toextendanddevelopthebookMIinallits
ChaptersI-VIII,andinthesequeltheywillbedescribedintheirrespectiveorderof
subjects.Inadditionanimportantnewareawasincludedaround2003:thedomain
ofprojectivelimitsandstochasticprocesses.Themainsurveyarticlesare(10),the
notesofaseriesoflecturesataworkshopin2001,and(20)fromtheGu¨nterLumer
Volumein2007.Furthersurveyarticlesare(11)onafundamentalideaofChoquet
connectedwith ChapterIV,and(18)onprojectivelimitsandstochasticprocesses.
For quite some time now it has been clear that the inner regular development is
muchmorefundamentalthantheouterone.
MI Chapters I-III: THE FOUNDATIONS. The most importantrelevant article is
(24)asnotedin thePreface.Itpresentsnewandmuchsimplerproofsofthebasic
results,totheextentneededintheentirework.Furtherimportantadditionsare(3)
PartIontheconnectionbetweenouterandinnerregularextensions,and(22)onthe
characterizationoftheuniquemaximalextensions.In(1)and(13)§3theformation
ofdirectandinverseimagesundermapsistreated,primarilyforpremeasures,and
restrictedtoinnerones.
MI Chapter IV: THE CHOQUET INTEGRAL. In MI and in the first articles, the
Choquet integral has been called the horizontal integral. The systematic and ex-
clusive use of this integral is one of the basic reasons for a comprehensive and
transparentdevelopment.The main new contributionsare (4) §1 and (12) devoted
toChoquet’sideaon(sub/super)additivefunctionalsquotedabove,withthesurvey
article(11).
MIChapterV:INTEGRAL REPRESENTATIONSOFISOTONE FUNCTIONALS.In
thispartofMIthecase•=(cid:2)turnedouttobeofaspecialandlessfundamentalkind.
Thereforethenewarticlesconcentrateon• = στ.Thebasicstepis(4),wherethe
representationtheoremsofMIareextendedtoouterandinneroneswithmuchwider
classes of domains, and of course in terms of the Choquetintegral. As before the
innerdevelopmentturnsouttobethesuperiorone.Thepaper(8)dealswithcertain
specialsituations.Thefinalstepisthen(19),whichintheinnersituationobtainsa
kindofuniversalDaniell-Stone-Riesztheorem.Weemphasizethatalltheseresults
ix
x Introduction
contain uniqueness assertions for the representative premeasures. In addition (3)
Part II and (5), and also the survey part (10) §§7-8, are devoted to a comparison
with the traditional Daniell-Stone and Bourbaki theories. The results are a clear
confirmationforthenewconcepts.
MI Chapter VI: TRANSPLANTATION THEOREMS. In the present contextthese
theorems are an importantdevice in order to transfer inner premeasures and their
descendantsfromonedomaintoanotherone.Thenewcontributionsare(7)interms
of upper envelopes, (13) §2 needed for projective limits, and (25) which pursues
ideasofDavidFremlinfortheconstructionofRadonmeasures.
MI Chapter VII: PRODUCTS. In the new context the formation of products is
completelynovel.Thefulltreatmentpresupposestheinnersituation,butsometimes
also outerformationsare needed.The treatmentin MI, for which we also referto
the survey part (10) §6, is restricted to two factors. The decisive point is that the
explicitproductconstructionisperformedonthelevelofinnerpremeasuresviathe
Choquet integral, and not on that of full measures. Then (2) treats the complete
situationsoffiniteandinfiniteproducts;for• = τ theinfinitecaseisundercertain
limitations.Atlastthearticle(21)returnstotheFubini-Tonellitheoremsandarrives
atcomprehensivebutyetplainversions.
MI Chapter VIII: APPLICATIONS. The most remarkable new article seems to
be (6): it unites several traditional concepts of signed contents and measures via
a new difference formation based on MI §23. Then (15) and (16) extend MI §24
andpresentnewversionsoftheRadon-Nikody´mandLebesguedecompositionthe-
orems.Thesecondoneisbasedon(6)andunitesthemultitudeofdifferentversions
of the resultinto a commonone, valid for arbitrary contentsandpresentedin ex-
plicitformulas.Atlast(9)isacontributiontothenotionofconicalmeasuresdueto
Choquet,basedonsublinearfunctionalsfromtheauthor’sworkinconvexanalysis.
PROJECTIVELIMITSANDSTOCHASTICPROCESSES.Thisisasequenceoffour
papersplusthe surveyarticle (18).Itstartsin (13)withversionsofthe Prokhorov
andKolmogorovprojectivelimittheoremsintermsofinner•probabilitypremea-
sures.Theincorporationofthecase • = τ achievesforthefirsttimethattheKol-
mogorovtheoremisabletoovercomethebarrierofso-calledcountablydetermined
subsetsinuncountableproductspaces:thetheoremproducesinnerτ premeasuresof
whichthemaximalprobabilitymeasureextensionscanhaveimmensedomains.In
(13)§6thereisafirstexample:oneprovesatoncethat,forthenewWienermeasure
onthepathspaceR[0,∞[,thesubsetofcontinuouspathsismeasurablewithfullmea-
sure.In(14)thenonepasses,afteracertainfortificationofthepresentKolmogorov
theorem,totheresultantconceptofstochasticprocesses.Thepaperpresentsforthe
firsttimethenowobviousadequatedefinitionfortheiressentialsubsetsinthepath
space.Thisfinishestheunfortunateanddoubtfulformerpracticeofusingtheouter
measuresoftheoldcanonicalprobabilitymeasures.Thesecondexampleafterthe
BrownianmotionisthenthePoissonprocess.
The paper (17) assumes a Polish state space with a certain local compactness
conditionandthetimedomain[0,∞[,andthenprovesforallstochasticprocesses
that certain subsets of the path space are measurable, that is in the domainof the