The Kurzweil-Henstock Integral for Undergraduates: A Promenade Along the Marvelous Theory of Integration

227 Pages·2018·3.24 MB·English

by  Alessandro Fonda

Compact Textbooks in Mathematics
Alessandro Fonda
The Kurzweil-
Henstock
Integral for
Undergraduates
A Promenade Along the Marvelous
Theory of Integration
CompactTextbooksinMathematics
CompactTextbooksinMathematics
This textbook series presents concise introductions to current topics in
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Moreinformationaboutthisseriesat:http://www.springer.com/series/11225
AlessandroFonda
The
Kurzweil-Henstock
Integral for
Undergraduates
A Promenade Along the Marvelous
Theory of Integration
AlessandroFonda
DipartimentodiMatematicaeGeoscienze
Universita`degliStudidiTrieste
Trieste,Italy
ISSN2296-4568 ISSN2296-455X (electronic)
CompactTextbooksinMathematics
ISBN978-3-319-95320-5 ISBN978-3-319-95321-2 (eBook)
https://doi.org/10.1007/978-3-319-95321-2
LibraryofCongressControlNumber:2018949935
MathematicsSubjectClassification(2010):26A39,26A42,26A63,26A66,26B15,26B20,28A75
©SpringerNatureSwitzerlandAG2018
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v
TheauthorwithJaroslavKurzweilinSeptember2008
To Sofia,Marcello,andElisa
ix
Introduction
This book is the outcome of the beginners’ courses held over the past
few years for my undergraduate students. The aim was to provide them
with a general and sufficiently easy to grasp theory of the integral. The
integral in question is indeed more general than Lebesgue’s in RN, but its
construction is rather simple, since it makes use of Riemann sums which,
beinggeometricallyviewable,areeasilyunderstandable.
Thisapproachtothetheoryoftheintegralwasdevelopedindependently
byJaroslavKurzweilandRalphHenstocksince1957(cf.[5,8]).Anumber
ofbooksarenowavailable[1,4,6,7,9–13,15–19,21].However,Ifeelthat
mostofthesemonographsareaddressedto anexpertreader, ratherthanto
a beginner student. This is why I wanted to maintain here the exposition
at a very didactical level, trying to avoid as much as possible unnecessary
technicalities.
Thebookisdividedintothreemainchaptersandfiveappendices,which
Inowbrieflydescribe,mainlyasaguideforthelecturer.
Thefirstchapteroutlinesthetheoryforfunctionsofonerealvariable.I
have done my best to keep the explanation as simple as can be, following
asfaraspossiblethelinesofthetheoryoftheRiemannintegral.However,
therearesomeinterestingpeculiarities.
▬ The FundamentalTheorem of differential and integral calculus is very
general and natural: one only has to assume the given function to be
primitivable, i.e., to be the derivative of a differentiable function. The
proofissimpleandclearlyshowsthelinkbetweendifferentiabilityand
integrability.
▬ The generalized integral, on a bounded but not compact interval, is
indeedastandardintegral:infact,Hake’stheoremshowsthatafunction
having a generalized integral on such an interval can be extended to a
function which is integrable in the standard sense on the closure of its
domain.
▬ IntegrablefunctionsaccordingtoLebesguearethosefunctionswhichare
integrableandwhoseabsolutevalueisintegrable,too.
Inthesecondchapter,thetheoryisextendedtorealfunctionsofseveral
real variables. No difficulties are encountered while considering functions
defined on rectangles. When the functions are defined on more general
domains, however, an obstacle arises concerning the property of additivity
on subdomains. It is then necessary to limit one’s attention to functions
which are integrable according to Lebesgue, after having introduced the
concept of measurable set. On the other hand, for the Fubini Reduction
TheoremthereisnoneedtodealwithLebesgueintegrablefunctions.Ithasa
rathertechnicalbutconceptuallysimpleproof,whichonlymakesuseofthe
x Introduction
Kurzweil–Henstockdefinition.In theTheorem on theChangeofVariables
intheintegral,onceagaincomplicationsmayarise(see,e.g.,[2]),sothatI
againdecidedtolimitthediscussiononlytofunctionswhichareintegrable
according to Lebesgue. The same goes for functions which are defined on
unbounded sets. These difficulties are intrinsic, not only at an expository
level,andresearchonsomeoftheseissuesisstillbeingcarriedout.
Thethirdchapterillustratesthetheoryofdifferentialforms.Theaimis
toprovetheclassicaltheoremscarryingthenameofStokes,andPoincaré’s
theorem on exact differential forms. Dimension 3 has been considered
closely:indeed,thetheoremsbyStokes–CartanandPoincaréareprovedin
this chapter onlyin this case, and thereader is referred to AppendixB for
the general proof. Also, I opted to discuss only thetheory for M-surfaces,
withoutgeneralizingandextendingittomorecomplexgeometricalobjects
(see however Appendix C). In some parts of this chapter, the regularity
assumptions could be weakened, but I did not want to enter into a topic
touchingastillongoingresearch.
In Appendix A, the basic facts about differential calculus in RN are
reviewed.
InAppendixB,thetheoremsbyStokes–CartanandPoincaréareproved.
Theproofsarerathertechnicalbutdonotpresentgreatconceptualdifficul-
ties.
In Appendix C, one can find a brief introduction to the theory of
differentiable manifolds, with particular emphasis on the corresponding
version of the Stokes–Cartan theorem. I did not want to deal with this
argumentextensively,andtheproofsareonlysketched.Foramorecomplete
treatment,wereferto[20].
InAppendixD,oneofthemostsurprisingresultsofmodernmathematics
is reported, the so-called Banach–Tarski paradox. It states that a three-
dimensional ball can be divided into a certain number of subsets which,
aftersomewell-chosenrotationsandtranslations,finallygivetwoidentical
copiesofthestartingball.Whyreportingonthisinabookaboutintegration?
Well, the Banach–Tarski paradox shows the existence of sets which are
notmeasurable(a rotation and a translation maintain the measureof a set,
providedthissetismeasurable!),anditdoesthisinaveryspectacularway.
AppendixEentailsashorthistoricalnoteontheevolutionoftheconcept
ofintegral.Thisnoteisbynomeanscomplete.Theaimistogiveanideaof
theroleplayedbytheRiemannsumsinthedifferentstagesofthehistoryof
theintegral.
Note ApreliminaryversionofthisbookwaspublishedinItalianunderthe
title Lezioni sulla teoria dell’integrale. It has been revised here, extending
andimprovingmostofthearguments.
xi
Contents
1 FunctionsofOneRealVariable................................. 1
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1.1 P-PartitionsandRiemannSums .................................... 1
1.2 TheNotionofδ-Fineness ........................................... 3
1.3 IntegrableFunctionsonaCompactInterval........................ 6
1.4 ElementaryPropertiesoftheIntegral .............................. 8
1.5 TheFundamentalTheorem......................................... 11
1.6 PrimitivableFunctions.............................................. 13
1.7 PrimitivationbyPartsandbySubstitution.......................... 18
1.8 TheCauchyCriterion ............................................... 23
1.9 IntegrabilityonSub-Intervals ...................................... 25
1.10 R-IntegrableFunctionsandContinuousFunctions.................. 28
1.11 TheSaks–HenstockTheorem....................................... 32
1.12 L-IntegrableFunctions ............................................. 35
1.13 TheMonotoneConvergenceTheorem.............................. 40
1.14 TheDominatedConvergenceTheorem............................. 45
1.15 IntegrationonNon-CompactIntervals ............................. 47
1.16 TheHakeTheorem ................................................. 53
1.17 IntegralsandSeries................................................. 56
2 FunctionsofSeveralRealVariables ............................ 61
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2.1 IntegrabilityonRectangles......................................... 61
2.2 IntegrabilityonaBoundedSet..................................... 66
2.3 TheMeasureofaBoundedSet ..................................... 68
2.4 TheChebyshevInequality.......................................... 72
2.5 NegligibleSets ..................................................... 73
2.6 ACharacterizationofBoundedMeasurableSets ................... 75
2.7 ContinuousFunctionsandL-IntegrableFunctions.................. 79
2.8 LimitsandDerivativesUndertheIntegrationSign.................. 82
2.9 TheReductionFormula............................................. 87
2.10 ChangeofVariablesintheIntegral................................. 96
2.11 ChangeofMeasurebyDiffeomorphisms ........................... 102
2.12 TheGeneralTheoremontheChangeofVariables.................. 104
2.13 SomeUsefulTransformationsinR2................................. 107
2.14 CylindricalandSphericalCoordinatesinR3 ........................ 110
2.15 TheIntegralonUnboundedSets ................................... 113
3 DifferentialForms ............................................... 125
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3.1 TheVectorSpaces(cid:3)M(RN)......................................... 125
3.2 DifferentialFormsinRN............................................ 127