Notes on Real Analysis and Measure Theory: Fine Properties of Real Sets and Functions

256 Pages·2022·3.458 MB·English

by  Alexander Kharazishvili

Springer Monographs in Mathematics
Editors-in-Chief
MinhyongKim,SchoolofMathematics,KoreaInstituteforAdvancedStudy,Seoul,
South Korea, InternationalCentreforMathematicalSciences,Edinburgh,UK
KatrinWendland,SchoolofMathematics,TrinityCollegeDublin,Dublin,Ireland
SeriesEditors
SheldonAxler,DepartmentofMathematics,SanFranciscoStateUniversity,
SanFrancisco,CA,USA
MarkBraverman,DepartmentofMathematics,PrincetonUniversity,Princeton,
NY,USA
MariaChudnovsky,DepartmentofMathematics,PrincetonUniversity,Princeton,
NY,USA
TadahisaFunaki,DepartmentofMathematics,UniversityofTokyo,Tokyo,Japan
IsabelleGallagher,DépartementdeMathématiquesetApplications,EcoleNormale
Supérieure,Paris,France
SinanGüntürk,CourantInstituteofMathematicalSciences,NewYorkUniversity,
NewYork,NY,USA
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PascalMassart,DépartementdeMathématiques,UniversitédeParis-Sud,Orsay,
France
AlbertoA.Pinto,DepartmentofMathematics,UniversityofPorto,Porto,Portugal
GabriellaPinzari,DepartmentofMathematics,UniversityofPadova,Padova,Italy
KenRibet,DepartmentofMathematics,UniversityofCalifornia,Berkeley,CA,
USA
RenéSchilling,InstituteforMathematicalStochastics,TechnicalUniversity
Dresden,Dresden,Germany
PanagiotisSouganidis,DepartmentofMathematics,UniversityofChicago,
Chicago,IL,USA
EndreSüli,MathematicalInstitute,UniversityofOxford,Oxford,UK
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IL,USA
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Alexander Kharazishvili
Notes on Real Analysis
and Measure Theory
Fine Properties of Real Sets and Functions
Alexander Kharazishvili
Andrea Razmadze Mathematical Institute
Tbilisi State University
Tbilisi, Georgia

ISSN 1439-7382 ISSN 2196-9922 (electronic)
Springer Monographs in Mathematics
ISBN 978-3-031-17032-4 ISBN 978-3-031-17033-1 (eBook)
https://doi.org/10.1007/978-3-031-17033-1

Mathematics Subject Classification (2020): 26-02, 28-02, 54-02, 03-02, 22-02

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Preface
Thisbookisconcernedwithacertaincircleofquestionsfromclassicalrealanalysis
andmeasuretheory.First,wewouldliketostressthatthechoiceoftopicspresented
inthetextiscompletelydictatedbyourresearchinterestsandpreferences.Neverthe-
less,wehopethatthismaterialwillbeofinteresttoa(moreorless)wideaudience
ofmathematicians,especiallytothoseworkinginvariousbranchesofmodernmath-
ematicalanalysis,probabilitytheory,thetheoryofstochasticprocesses,andgeneral
topology.Wealsotouchuponset-theoreticalaspectsofthemesdiscussedinthebook.
Perhaps,someoftheseideaswillbeinterestingforset-theoristsandwillhelpthem
to find new applications of their methods to various problems of real analysis and
measuretheory.
Itshouldbenotedthatseveralquestionsconsideredinthisbookarecloselyrelated
totopicscoveredintheauthor’smonograph:
(*) TopicsinMeasureTheoryandRealAnalysis.AtlantisPress,Paris(2009).
Insomerespects,thepresentbookmaybetreatedasacontinuationof(*),atleast
in the sense of their common ideology and methods. However, the material given
belowissubstantiallyself-contained,sopotentialreaderscanstudyitindependently
of(*).Weonlysupposethatapotentialreaderissomewhatfamiliarwiththebasic
conceptsofsettheory,realanalysisandmeasuretheory,generaltopologyandgroup
theory.Allconceptsfromtheabove-mentionedfieldsofmathematicsthatareused
in the text can be found in standard textbooks and monographs. In this context, it
makes sense to recall several well-known books which will probably be useful for
potentialreadersofthepresentmanuscript.
• K. Hrbacek and T. Jech. Introduction to Set Theory. Marcel Dekker, New York
(1999).
• D.H. Fremlin. Consequences of Martin’s Axiom. Cambridge University Press,
Cambridge(2011).
• H.L.RoydenandP.M.Fitzpatrick.RealAnalysis.ChinaMachinePress,Beijing
(2010).
• E.M. Stein and R. Shakarchi. Real Analysis: Measure Theory, Integration, and
HilbertSpaces.PrincetonUniversityPress,Princeton(2005).
v
vi Preface
• V.Bogachev.MeasureTheory.Springer-Verlag,Berlin-Heidelberg(2007).
• K.Kuratowski.Topology,vol.1.AcademicPress,London-NewYork(1966).
• R.Engelking.GeneralTopology.PWN,Warszawa(1985).
• A.G.Kurosh.Gruppentheorie,2vols.AkademieVerlag,Berlin(1970,1972).
Ofcourse,theabovetextscovermuchmoreextensivematerialaboutsettheory,real
analysisandmeasuretheory,generaltopologyandgrouptheory.
Fortheconvenienceofthereader,wewouldliketodescribe(brieflyandschemat-
ically)thethemeswhicharediscussedinthisbook.
InChapter1generalpropertiesofsemicontinuousreal-valuedfunctionsarecon-
sidered which are useful for the study of various topics in mathematical analysis.
Sometypicaltheoremsandfactsconcerningsemicontinuousfunctionsarepresented
(includingaseparationprincipleforsuchfunctions).
Chapter2isdevotedtotheoscillationsofreal-valuedfunctions.If 𝐸 isatopo-
logical space and R denotes the usual real line, then, for any function 𝑓 : 𝐸 → R
whichislocallyboundedon 𝐸,itmakessensetointroduceitsoscillationfunction
𝑂𝑓 : 𝐸 →Rdefinedasfollows:
𝑂𝑓(𝑥) =limsup𝑦→𝑥𝑓(𝑦)−liminf𝑦→𝑥𝑓(𝑦) (𝑥 ∈ 𝐸).
It turns out (and it is not hard to show) that 𝑂𝑓 is always a non-negative and
upper semicontinuous function. So the natural question arises whether every non-
negativeanduppersemicontinuousreal-valuedfunctionon 𝐸 canbetreatedasan
oscillationfunction.Theanswertothisquestiondependsonstructuralpropertiesof
theinitialspace 𝐸.Westudythequestionforcertainclassesoftopologicalspaces.
Inparticular,itisdemonstratedthattheanswerispositiveformanystandardspaces
ofmathematicalanalysis(e.g.,for𝐸 =R).
InChapter3wedealwiththoserestrictionsofareal-valuedfunction 𝑓 whichhave
relativelyniceproperties,e.g.,arecontinuousormeasurableonanon-smallsubset
ofthedomainof 𝑓 (ofcourse,herethenotionof“smallness”maybeinterpretedin
different ways). Typical results in this direction are, for example, Luzin’s theorem
on the 𝐶-property of Lebesgue measurable functions, Blumberg’s theorem on the
existenceofacontinuousrestrictiontoaneverywheredensesubsetofthedomain,
and the theorem on the existence of a monotone restriction to a nonempty perfect
subsetofthedomain.Wepresentsomecounterexamplestotheoremsoftheabove-
mentionedkind.Forinstance,itisshownthatthereexistsacontinuousreal-valued
function on the interval [0,1] which does not admit a monotone restriction to a
Lebesguemeasurablesubsetof[0,1]havingstrictlypositivemeasure.Besides,itis
indicatedthatnoSuslinlinesatisfiestheanalogofBlumberg’stheorem[14].Inthe
samechapterthenotionofaso-calledmagicsetisalsoconsideredanditisproved,
by using some additional set-theoretical assumptions, that magic sets exist. In this
context,itshouldbenotedthatincertainmodelsofZFCtheorysuchsetsareabsent
(see[26]).
Preface vii
Chapter 4 is devoted to the behavior of certain kinds of small sets (Luzin sets,
Sierpiński sets, universal measure zero sets, etc.) under bijective continuous map-
pings.Itisdemonstratedthat,ingeneral,theimageofasmallsetunderamappingof
theabove-mentionedtypecanbeofapathologicalnature.Forexample,amongsuch
imagesonemayencounteraBernsteinset,whichisabsolutelynonmeasurablewith
respecttotheclassofthecompletionsofallnonzero𝜎-finitediffuseBorelmeasures
onuncountablePolishtopologicalspaces.Someotherpathologiesrelatedtothejust
describedsetsarealsodiscussedinthesamechapter.
Chapter5isconcernedwithdescriptivepropertiesofreal-valuedfunctions.The
central question raised in this chapter can be formulated as follows: is it possible
to obtain (at least, in certain models of ZFC theory) an absolutely nonmeasurable
function𝜙:R→Rwhosedescriptivestructureisrelativelygood.Forinstance,one
mayrequirethatthegraphof𝜙mustbelongtotheprojectivehierarchyofLuzinand
Sierpiński.Theanswertothisquestionturnsouttobepositive.Indeed,weshowthat
ifthereallineRadmitsaprojectivewell-orderingisomorphictotheleastuncountable
ordinal𝜔 ,thenthereexistsanabsolutelynonmeasurablefunction𝜙:R→Rwhose
1
graphisaprojectivesubsetoftheEuclideanplaneR2.Hereacentralroleisplayedby
Luzinsetsandthecharacterizationofabsolutelynonmeasurablefunctionsinterms
ofuniversalmeasurezerosetsandpre-imagesofpoints.Itshouldbeespeciallynoted
thatananalogousresultcanalsobeobtainedunderMartin’sAxiom(MA),whichis
muchweakerthantheContinuumHypothesis(CH).
ItiswellknownfromclassicaldescriptivesettheorythatanytwoBorelsubsets
ofaPolishtopologicalspaceareBorelisomorphicifandonlyiftheyhavethesame
cardinality(see,e.g.,[111]).AfterBorelsets,theso-calledanalytic(orSuslin)sets
areofgreatimportanceintheprojectivehierarchyofLuzinandSierpiński,andthe
naturalquestionariseswhetheraresultanalogoustotheoneformulatedaboveholds
true for analytic subsets of a Polish space. In Chapter 6 we present a remarkable
theoremofA.MaitraandCz.Ryll-Nardzewski[121]statingthatincertainmodels
ofZFCtheorytherearetwouncountableanalyticsubsetsofthereallineRwhich
arenotBorelisomorphic.Inparticular,ifGödel’sConstructibilityAxiom(V=L)
holds, then one can assert that there exist two proper analytic sets in R which are
not Borel isomorphic. To establish this profound result, one needs several delicate
auxiliaryfactsfromdescriptivesettheory,concerningthestructureofconstituents
ofco-analyticsubsetsofR.
ThematerialofChapter7isfocusedoniteratedintegralsofreal-valuedfunctions
of two real variables. After the classical example of Sierpiński (see [155, 157]),
whichwasconstructedwiththeaidoftheContinuumHypothesis,itbecameknown
that,forafunction 𝑓 : [0,1]2 → [0,1],bothiteratedintegralsof 𝑓 mayexist,but
their values may differ from each other. So, the equality of these iterated integrals
needs additional regularity assumptions on 𝑓. In this connection, it makes sense
to study the vector space F of those real-valued functions 𝑓 on [0,1]2 for which
bothiteratedintegralsexistandcoincide.Followingtheoldandlessknownworkof
Pkhakadze[142],weconsidersomepropertiesofthisvectorspace.Inparticular,we
showthatifCHholds,thenF isnotanalgebra,i.e.,therearetwofunctions 𝑓 ∈ F
1
viii Preface
and 𝑓 ∈ F suchthat 𝑓 · 𝑓 ∉ F,andwealsoshowthatF isnotalattice,i.e.,there
2 1 2
existtwofunctions𝑔 ∈ F and𝑔 ∈ F suchthatmax(𝑔 ,𝑔 ) ∉ F.Severalrelated
1 2 1 2
resultsarediscussedinthesamechapterandinaccompanyingexercises.
InChapter8,motivatedbyawell-knowntheoremofSteinhaus[163],weconsider
severalversionsoftheSteinhauspropertyforaninvariantmeasure𝜇,fromthepoint
ofviewofdensitypointsof 𝜇-measurablesets(inthestandardmeasure-theoretical
sense).Heresomestatementsareprovedwhichindicatecloseconnectionsbetween
the concepts of ergodicity, the Steinhaus property, and 1/2-density points. The
chapter culminates with a theorem stating the existence of an ergodic translation-
invariantmeasure𝜈 onRwhichextendstheLebesguemeasureonRandforwhich
thereisa𝜈-measurableset 𝑋 with𝜈(𝑋) >0whosedifferenceset
𝑋−𝑋 = {𝑥−𝑥′ : 𝑥 ∈ 𝑋, 𝑥′ ∈ 𝑋}
istotallyimperfectinRandat 𝜇-almostallpointsof 𝑋 thedensityof 𝑋 isgreater
thanorequalto1/2.
Chapter 9 covers several questions closely related to the measurability problem
of selectorsassociated with subgroupsof the classicaladditive group (R,+). Here
two cases should be distinguished: the case ofa countable subgroup of (R,+) and
the case of an uncountable subgroup of (R,+). Our primary focus is on the case
ofaninfinitecountablegroup𝐺 ⊂ Rand,inaddition,weconsidertheproblemof
measurability not only for 𝐺-selectors but also for partial 𝐺-selectors. It turns out
thatthesituationswith𝐺-selectorsandpartial𝐺-selectorsareessentiallydifferent.
InthesamechapterwegiveapositiveanswertoaproblemofG.Lazouconcerning
Vitali–BernsteinselectorsandpartialVitali–Bernsteinselectors.
InChapter10,weprimarilyinvestigatetheclassofso-callednegligiblesubsetsof
agivengroundspace𝐸equippedwithatransformationgroup𝐺.Brieflyspeaking,a
negligiblesubsetof𝐸 (withrespectto𝐺)isanyset𝑋 ⊂ 𝐸 whichisofmeasurezero
forsomenonzero 𝜎-finite𝐺-invariant(𝐺-quasi-invariant)measureon 𝐸,andalso
is of measure zero with respect to every 𝜎-finite 𝐺-invariant (𝐺-quasi-invariant)
measure 𝜇 on 𝐸 forwhich 𝑋 ∈ dom(𝜇).Themainresultformulatedandprovedin
thischapteristhetheoremstatingthatanarbitraryuncountablesolvablegroup(𝐺,·)
admitsadecomposition{𝑋,𝑌,𝑍}intothreedisjointnegligiblesetswithrespectto
thegroupofalllefttranslationsof𝐺.Asatrivialconsequenceofthisfact,weobtain
that, for any nonzero 𝜎-finite 𝐺-quasi-invariant measure 𝜈 on (𝐺,·), there exist at
leasttwosetsfrom{𝑋,𝑌,𝑍}whicharenonmeasurablewithrespectto𝜈.Notethat,
toobtaintheseresultsacertaingeometricformofCH,statedbySierpiński[155],
turnedouttobeveryhelpful.
Chapter11isdevotedtosomerelationshipsbetweennegligiblesetsandabsolutely
nonmeasurable sets in uncountable commutative (more generally, in uncountable
solvable)groups.Inparticular,itisdemonstratedthat,foranyuncountablesolvable
group (𝐺,·), there exist two 𝐺-negligible subsets 𝐴 and 𝐵 of 𝐺 such that 𝐴∪ 𝐵
isa𝐺-absolutelynonmeasurablesubsetof𝐺.Thisresultshows,inparticular,that
Preface ix
theunionoftwonegligiblesetscanbeextremelybadfromthemeasure-theoretical
viewpoint.
Chapter 12 is concerned with measurability properties of Mazurkiewicz sets.
Recallthat 𝑋 ⊂ R2 isaMazurkiewiczsetifcard(𝑋 ∩𝑙) = 2foreverystraightline
𝑙 ofR2.Suchaparadoxicalset 𝑋 wasfirstconstructedbyS.Mazurkiewicz[125]in
1914,withtheaidofthemethodoftransfiniterecursion.Descriptivepropertiesof
Mazurkiewicz type subsets of the plane may be absolutely different. For instance,
thefollowingfactsarewell-known:
(i) thereexistsanowheredenseMazurkiewiczsetwhichisofLebesguemeasure
zero;
(ii) thereexistsaMazurkiewiczsetwhichintersectseachBorelsubsetofR2 with
strictlypositiveLebesguemeasure;
(iii) ifaMazurkiewiczsetisananalyticsubsetofR2,thenitisaBorelsubsetofR2.
Also,ashasrecentlybeenproved,inGödel’smodelLofZFCtheorythereexistsa
co-analyticMazurkiewiczsubsetofR2(see[127,128]).Ontheotherhand,itisstill
unknownwhetherthereexistsaMazurkiewiczsetwhichisaBorelsubsetofR2.In
thischapterweconsiderthebehaviorofMazurkiewiczsetswithrespecttotheclass
M′(R2) ofallnonzero𝜎-finitetranslation-invariantmeasuresonR2.Inparticular,
weshowthateveryMazurkiewiczsetisnegligiblewithrespecttoM′(R2)andthat
thereareMazurkiewiczsetswhichareabsolutelynegligiblewithrespecttoM′(R2).
Moreover,itisprovedinthesamechapterthatthereexistMazurkiewiczsetswhich
arenotabsolutelynegligiblewithrespecttoM′(R2).
In Chapter 13 we use Marczewski’s method of extending nonzero 𝜎-finite in-
variant (quasi-invariant) measures and evaluate the number of possible extensions
of such measures in the special case when the role of a ground set 𝐸 is played by
thereallineRandtheroleofatransformationgroup𝐺 isplayedbythegroupofall
translationsofR.Wealsotouchuponsomenonseparableextensionsofmeasureson
R.
AppendixAisofapurelyset-theoreticalflavor.Accordingtotheclassicaldefini-
tionofBolzanoandDedekind,aset𝐸 isinfinite(moreprecisely,𝐷-infinite)ifthere
existsabijectivemappingof𝐸ontoitspropersubset.Inotherwords,𝐸is𝐷-infinite
if and only if 𝐸 is equinumerous with some proper subset of 𝐸. Throughout the
book, we are dealing with uncountable ground (base) sets equipped with various
transformationgroups.Motivatedbythiscircumstance,wediscussinthisappendix
an analogous abstract characterization of uncountable sets in terms of their self-
mappings.Inparticular,itisprovedthataset𝐸 isuncountableifandonlyif,forany
mapping 𝑓 : 𝐸 → 𝐸,thereexistsapropersubset𝑍of𝐸suchthatcard(𝑍) =card(𝐸)
and 𝑓(𝑍) ⊂ 𝑍. Several related questions concerning the fundamental mathemati-
cal concepts of finiteness (countability) and infiniteness (uncountability) are also
touchedupon.