The Integral: An Easy Approach after Kurzweil and Henstock 9780521779685, 0521779685

Table of contents :
Preface
List of Symbols
1. lntroduction
1.1 Historical remarks
1.2 Notation and the Riemann definition
1.3 Basic theorems, upper and lower integrals
1.4 Differentiability , continuity and integrability
1.5 Limit and R-integration
1.6 Exercises
2. Basic Theory
2.1 Introduction
2.2 Motivation
2.3 Cousin's lemma
2.3.1 Applications of Cousin's lemma
2.4 The definition
2.5 Basic theorems
2.6 The Fundamental Theorem of calculus
2.7 Consequences of the Fundamental Theorem
2.8 Improper integrals
2.9 Integrals over unbounded intervals
2.10 Alternative approach to integration over unbouuded intervals
2.11 Negligible sets
2.12 Complex valued function
2.13 Exercises
3. Development of the Theory
3.1 Equivalent forms of the definition
3.2 Henstock's lemma
3.3 Functions of bounded variation
3.4 Absolute integrability
3.5 Limit and KH-integration
3.6 Absolute continuity
3.7 Equiintegrability
3.8 Differentiation of integrals
3.9 Characterization of the KH-integral
3.10 Lebesgue points, approximation by step functions
3.11 Measurable functions and sets
3.11.1 A non-measurable set
3.12 The McShane integral
3.12.1 A short proof
3.13 The Lebesgue integral
3.13.1 F. Riesz' definition
3.13.2 Quick proofs
3.14 Differentiation almost everywhere
3.15 Exercises
4. The SL-integral
4.1 The strong Luzin condition
4.2 SL-integration
4.3 Limit and SL-integration
4.4 Equivalence with the KH-integral
4.5 Exercises
5. Generalized AC Functions
5.1 Prologue
5.2 Uniformly AC functions
5.3 AC* and VB* on a set
5.4 ACG* functions
5.5 Controlled convergence
5.6 Exercise
6. Integration in Several Dimensions
6.1 Introduction
6.1.1 Sets in R^n
6.2 Divisions, partitions
6.3 The definition
6.4 Basic theorems
6.4.1 Prelude to Fubini's theorem
6.5 Other theorems in R^n
6.5.1 Negligible sets
6.5.2 Henstock's lemma
6.5.3 Absolute integrability
6.5.4 Convergence, measurability, AC
6.6 The Fubini theorem
6.7 Change of variables
6.7.1 Introductory examples
6.7.2 Notation, lemmas
6.7.3 The theorem
6.8 Exercises
7. Some Applications
7.1 lntroduction
7.2 A line integral
7.2.1 Green's theorem
7.2.2 The Cauchy theorem
7.3 Differentiation of series
7.4 Dirichlet's problem and the Poisson integral
7.5 Summability of Fourier series
7.6 Fourier series and the space L^2
7.7 Exercises
Appendix. Supplements
A.1 The Cantor set
A.1.1 An uncountable set
A.1.2 Cantor's discontinuum
A.2 Dini's Theorem
A.3 Sets in R^n
A.4 Uniqueness of the Dirichlet problem
A.5 Abel's theorem
A.6 Proof of Lemma 6.7.4
Bibliography
1-17
18-43
44-48
Index

Citation preview

AUSTRALIAN _M.ATHB?.1ATTCAL SOCIETY LECTUfl;F, SERJhlS Editor-in-Chief: Professor J.H. Loxton, School of Mathematics, P l,ys(cs, Cotnputing and Electronics, ~lac.qua,ie University, NSvV 2109, Australia Editors: Professor C .C. lleyde, School of t-iat hematical Sciences, Australian Xational Unlversit)', Canberra, ACT 0200, Australia Associate Profes$or ',\I .D. Neumann , Department of Mathen1atics, University of Melbourne, Parkville, Victoria 3052, Aoslralia Associate Professor C.E-.M Pearce, Department of Appl,cd Mathematics, University of Adelaide, SA 5005, Austral ia

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The Mathematics of Projectiles in Sport, N. DE MESTRE

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9

Algebra,r eigl,ty years ago Lebesgue ('1875-19~1) gave another l'vfax(l/(1,j)I, .f =< l , 2, . . . ,n). Lrit :e E [a , b] b;, arbitrary; it lies in somf: T u"'. ,,.,j and in equality ( J.8)

n J.t

7 lt,l11,duct·ion

for the partition ( 1.3) w itl, ~,

= n;

(j =fa a) and {a= a: yields n

IJ(:,;)(v" - •1,,,.)I < IAl I l

+

L

lf(u,) (1,, - 4,),

i = l ,i¢n

;\Dd consequently

l/ (x)I < nUAI + ~~ ~(o - a)). Since a, was arbi l n-,;ry t his shows f is bounded. Cnlcss someth ing is specifk-d to the contnLCy we slwJI assu,no for tho rHsL of Lhis chapter th at all ft111ctions appearing are bounded. t

1.3 Basic t h eorems, upper and lower integrals For a division D a.. &~ven in (1.1} we in(todLlce t.he Darboux Upper and lower suws S(D) and s(D) defule

tind a pusitlvc 6

I: f (O(v-u)- Al< 1· •

On each JnLcrval [u , 11! we choose ( so that

e

rn + 4(b -

a ) > /((.),

,;/?..

Similarly S(/J,) < A + e/2 aud ii1e4uAlity (ii}. T his (1.15) follows. To complete the proor i~ s11ffices to sh:,w (iv) • impLicatiou becornes obvious with 1.hc follo\\ing lemma.

It fol[Qwi; LhaL s(D") > A -

'*

LE rvHvlA 1.3.2 Foi· e11ery 1>0.1itive e t,here exists a. positive 6 su.ch thal.

S(D) -

1~•

-,, !. J •

J - s(D)

< e,

(1.1 6)

< "•

(1.17)

tuhenever n( D) < 6.

(1.18}

J:'roof Only (1.16) nreds proof since (1. 17} follows from ( 1. 16) applied to - f. For the proof of ( 1.Hi) we cionotc by /1 the valu~ of t he upper iutel','l'a l on