Measure and Integration: Publications 1997-2011 PDF
Preview Measure and Integration: Publications 1997-2011
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
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