Problems in Mathematical Analysis HI Integration

http://dx.doi.org/10.1090/stml/021

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STUDENT MATHEMATICAL LIBRARY Volume 21

Problems in Mathematical Analysis III Integration

W. J. Kaczor M.T. Nowak

#AMS AMERICAN MATHEMATICA L SOCIET Y

Editorial Boar d

David Bressoud , Chai r Danie l L . GorofT Car l Pomeranc e

2000 Mathematics Subject Classification. Primar y 00A07 , 26A42 ; Secondary 26A45 , 26A46 , 26D15 , 28A12 .

For additiona l informatio n an d updates o n this book , visi t www.ams.org/bookpages/stml-21

Library o f Congres s Cataloging-in-Publicatio n Dat a

Kaczor, W . J . (Wieslaw a J.) , 1949 -[Zadania z analizy matematycznej . English ] Problems i n mathematica l analysis . I . Rea l numbers , sequence s an d serie s /

W. J . Kaczor , M . T. Nowak . p. cm . — (Studen t mathematica l library , ISS N 1520-912 1 ; v. 4)

Includes bibliographica l references . ISBN 0-8218-2050- 8 (softcove r ; alk. paper ) 1. Mathematica l analysis . I . Nowak , M . T . (Mari a T.) , 1951 - II . Title .

III. Series .

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Contents

Preface vi i

Part 1 . Problem s

Chapter 1 . Th e Riemann-Stieltje s Integra l 3

§1.1. Propertie s o f the Riemann-Stieltje s Integra l 3

§1.2. Function s o f Bounded Variatio n 1 0

§1.3. Furthe r Propertie s o f the Riemann-Stieltje s Integra l 1 5

§1.4. Prope r Integral s 2 1

§1.5. Imprope r Integral s 2 8

§1.6. Integra l Inequalitie s 4 2

§1.7. Jorda n Measur e 5 2

Chapter 2 . Th e Lebesgu e Integra l 5 9

§2.1. Lebesgu e Measur e o n th e Rea l Lin e 5 9

§2.2. Lebesgu e Measurabl e Function s 6 6

§2.3. Lebesgu e Integratio n 7 1

§2.4. Absolut e Continuity , Differentiatio n an d Integratio n 7 9

VI Contents

§2.5. Fourie r Serie s 8 4

Part 2 . Solution s

Chapter 1 . Th e Riemann-Stieltje s Integra l 9 7

§1.1. Propertie s o f the Riemann-Stieltje s Integra l 9 7

§1.2. Function s o f Bounded Variatio n 11 4

§1.3. Furthe r Propertie s o f the Riemann-Stieltje s Integra l 12 6

§1.4. Prope r Integral s 14 3

§1.5. Imprope r Integral s 16 4

§1.6. Integra l Inequalitie s 20 7

§1.7. Jorda n Measur e 22 8

Chapter 2 . Th e Lebesgu e Integra l 24 7

§2.1. Lebesgu e Measur e o n the Rea l Lin e 24 7

§2.2. Lebesgu e Measurabl e Function s 26 8

§2.3. Lebesgu e Integratio n 28 1

§2.4. Absolut e Continuity , Differentiatio n an d Integratio n 29 6

§2.5. Fourie r Serie s 31 6

Bibliography - Books 35 1

Index 35 5

Preface

This i s a seque l t o ou r book s Problems in Mathematical Analysis I, II (Volume s 4 an d 1 2 i n th e Studen t Mathematica l Librar y series) . The book deals with the Riemann-Stieltje s integra l an d th e Lebesgu e integral fo r rea l function s o f one rea l variable . Th e boo k i s organize d in a way similar t o tha t o f the firs t tw o volumes , tha t is , i t i s divided into tw o parts : problem s an d thei r solutions . Eac h sectio n start s with a numbe r o f problems tha t ar e moderat e i n difficulty , bu t som e of the problem s ar e actuall y theorems . Thu s i t i s no t a typica l prob -lem book , bu t rathe r a supplemen t t o undergraduat e an d graduat e textbooks i n mathematica l analysis . W e hope tha t thi s boo k wil l b e of interes t t o undergraduat e students , graduat e students , instructor s and researche s in mathematical analysi s and it s applications . W e also hope tha t i t wil l be suitabl e fo r independen t study .

The first chapte r of the book is devoted to Riemann and Riemann -Stieltjes integrals . I n Sectio n 1. 1 w e conside r th e Riemann-Stieltje s integral wit h respec t t o monotoni c functions , an d i n Sectio n 1. 3 w e turn t o integratio n wit h respec t t o function s o f bounde d variation . In Sectio n 1. 6 w e collec t famou s an d no t s o famous integra l inequal -ities. Amon g others , on e ca n fin d OpiaP s inequalit y an d Steffensen' s inequality. W e clos e th e chapte r wit h th e sectio n entitle d "Jorda n measure". Th e Jorda n measure , als o called conten t b y some authors ,

vn

Vlll Preface

is no t a measur e i n th e usua l sens e becaus e i t i s no t coun t ably addi -tive. However , i t i s closely connecte d wit h th e Rieman n integral , an d we hope that thi s section will give the student a deeper understandin g of the idea s underlyin g th e calculus .

Chapter 2 deals with the Lebesgue measure an d integration . Sec -tion 2. 3 present s man y problem s connecte d wit h convergenc e theo -rems tha t permi t th e interchang e o f limi t an d integral ; LP spaces o n finite interval s ar e als o considered here . I n th e nex t section , absolut e continuity an d the relation between differentiation an d integration ar e discussed. W e present a proof o f the theore m o f Banac h an d Zareck i which state s tha t a function / i s absolutely continuou s o n a finite in -terval [a , b] i f and onl y i f it i s continuous an d o f bounded variatio n o n [a, b], an d map s set s o f measur e zer o int o set s o f measur e zero . Fur -ther, th e concep t o f approximate continuit y i s introduced. I t i s worth noting her e that ther e i s a certain analog y betwee n tw o relationships : the relationshi p betwee n Rieman n integrabilit y an d continuity , o n the on e hand , an d th e relationshi p betwee n approximat e continuit y and Lebesgu e integrability , o n th e othe r hand . Namely , a bounde d function o n [a , b] i s Riemann integrabl e i f an d onl y i f i t i s almost ev -erywhere continuous ; an d similarly , a bounde d functio n o n [a , b] is measurable, an d s o Lebesgu e integrable , i f an d onl y i f i t i s almos t everywhere approximatel y continuous . Th e las t sectio n i s devoted t o the Fourie r series . Give n th e existenc e o f extensiv e literatur e o n th e subject, e.g. , th e book s b y A . Zygmun d "Trigonometri c Series" , b y N. K . Bar i " A Treatis e o n Trigonometri c Series" , an d b y R . E . Ed -wards "Fourie r Series" , w e foun d i t difficul t t o decid e wha t materia l to includ e i n a boo k whic h i s primaril y addresse d t o undergraduat e students. Consequently , w e have mainl y concentrate d o n Fourie r co -efficients o f function s fro m variou s classe s an d o n basi c theorem s fo r convergence o f Fourie r series .

All the notation an d definition s use d i n this volume are standard . One can find them i n the textbook s [27 ] and [28] , which als o provid e the reade r wit h th e sufficien t theoretica l background . However , t o avoid ambiguit y an d t o make the boo k self-containe d w e start almos t every sectio n wit h a n introductor y paragrap h containin g basi c defi -nitions an d theorem s use d i n th e section . Ou r referenc e convention s

Preface IX

are bes t explaine d b y th e followin g examples : 1.2.1 3 or I , 1.2.1 3 or II, 1.2.13 , whic h denot e th e numbe r o f th e proble m i n thi s volume , in Volum e I o r i n Volum e II , respectively . W e als o us e notatio n an d terminology give n i n th e first tw o volumes .

Many problem s hav e been borrowe d freel y fro m proble m section s of journals lik e the America n Mathematica l Monthl y an d Mathemat -ics Today (Russian) , an d fro m variou s textbooks an d proble m books ; of thos e onl y book s ar e liste d i n th e bibliography . W e woul d lik e t o add tha t man y problem s i n Sectio n 1. 5 com e from th e boo k o f Ficht -enholz [10 ] an d Sectio n 1. 7 i s influence d b y th e boo k o f Rogosinsk i [26]. Regrettably , i t wa s beyon d ou r scop e t o trac e al l th e origina l sources, and w e offer ou r sincer e apologies i f we have overlooked som e contributions.

Finally, w e would lik e t o than k severa l peopl e fro m th e Depart -ment o f Mathematics o f Maria Curie-Sklodowsk a Universit y t o who m we are indebted . Specia l mentio n shoul d b e made o f Tadeusz Kuczu -mow an d Witol d Rzymowsk i fo r suggestion s o f severa l problem s an d solutions, an d o f Stanisla w Pru s fo r hi s counselin g an d Te X support . Words o f gratitud e g o t o Richar d J . Libera , Universit y o f Delaware , for hi s generou s hel p wit h Englis h an d th e presentatio n o f th e ma -terial. W e ar e ver y gratefu l t o Jadwig a Zygmun t fro m th e Catholi c University o f Lublin , wh o ha s draw n al l th e figure s an d helpe d u s with incorporatin g the m int o th e text . W e than k ou r student s wh o helped u s i n th e lon g an d tediou s proces s o f proofreading . Specia l thanks g o t o Pawe l Sobolewsk i an d Przemysla w Widelski , wh o hav e read th e manuscrip t wit h much care and thought , an d provide d man y useful suggestions . Withou t thei r assistanc e som e errors , no t onl y ty -pographical, coul d have passed unnoticed . However , we do accept ful l responsibility fo r an y mistake s o r blunder s tha t remain . W e woul d like to tak e thi s opportunit y t o than k th e staf f a t th e AM S fo r thei r long-lasting cooperation , patienc e an d encouragement .

W. J . Kaczor , M . T . Nowa k

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Bibliography - Book s

[1] T . M . Apostol , Mathematical Analysis, Addison-Wesle y Publishin g Company, Inc. , Reading , Mass. , 1957 .

[2] N . K . Bari , A treatise on trigonometric series. Vols. I, II, Pergamo n Press, Oxford , an d Macmillan , New York , 1964 .

[3] P . Biler , A . Witkowski , Problems in Mathematical Analysis, Marce l Dekker, Inc. , Ne w Yor k an d Basel , 1990 .

[4] F . Burk , Lebesgue Measure and Integration, A Wiley-Interscienc e Publication. Joh n Wile y & Sons , Inc. , New York , 1998 .

[5] D . L. Cohn, Measure Theory, Birkhauser , Boston , Basel , Berlin , 1993 .

[6] A . J . Dorogovcev , Matematiceskij analiz. Spravocnoe posobe, Vyscaj a Skola, Kiev , 1985 .

[7] A . J . Dorogovcev , Matematiceskij analiz. Sbornik zadac, Vyscaj a Skola, Kiev , 1987 .

[8] R . E . Edwards , Fourier series. Vol. I., Holt , Rinehar t an d Winston , Inc., New York , Montreal , London , 1967 .

[9] L . C . Evans , R . E . Gariepy , Measure Theory and Fine Properties of Functions, RC C Press , Boc a Rato n Fla. , New York , London , Tokyo , 1999.

[10] G . M . Fichtenholz , Differential-und Integralrechnung, I, II, III, V.E.B. Deutsche r Verla g Wiss. , Berlin , 1966-1968 .

[11] B . R. Gelbaum , J . M . H. Olmsted, Theorems and Counterexamples in Mathematics, Springer-Verlag , Ne w York , Berlin , Heidelberg , 1990 .

[12] G . H . Hardy , J . E . Littlewood , G . Polya , Inequalities, Cambridg e University Press , 1963 .

351

352 Bibliography - Book s

E. Hewitt , K . Stromberg , Real and Abstract Analysis, Springer -Verlag, Ne w York , Berlin , Heidelberg , 1965 .

E. W . Hobson , The Theory of Functions of a Real Variable. Vols. I, II, Dove r Publications , Inc. , Ne w York , 1958 .

W. J . Kaczor , M . T . Nowak , Problems in Mathematical Analysis I. Real Numbers, Sequences and Series, America n Mathematica l Soci -ety, Providence , RI , 2000 .

W. J . Kaczor , M . T . Nowak , Problems in Mathematical Analysis II. Continuity and Differentiation, America n Mathematica l Society , Providence, RI , 2001.

G. Klambauer , Mathematical Analysis, Marce l Dekker , Inc. , Ne w York, 1975 .

G. Klambauer , Real Analysis, America n Elsevie r Publishin g Com -pany, Inc. , Ne w Yor k Londo n Amsterdam , 1973 .

S. Lojasiewicz , An Introduction to the Theory of Real Functions, A Wiley-Interscience Publication , Joh n Wiley & Sons, Ltd. , Chichester , 1988.

D. S . Mitrinovic , Analytic Inequalities, Springer-Verlag , Ne w York , Berlin, Heidelberg , 1970 .

M. E . Munroe , Introduction to Measure and Integration, Addison -Wesley Publishin g Company , Inc. , Cambridge , Mass. , 1953 .

I. P . Natanson , Teoria funkcij vescestvennoj peremennoj. (Russian) , Gosudarstv. Izdat . Tehn.-Teoret . Lit. , Moscow , 1950 ; Englis h trans. , Vols. 1,2 , Ungar , Ne w York , 1955 , 1961.

J. Niewiarowski , Zadania z teorii miary, (Polish) , Wydawnictwo Uni -wersytetu Lodzkiego , Lodz , 1999 .

Yu. S . Ocan , Sbornik zadac po matematiceskomu analizu, (Russian) , Prosvesenie, Moskva , 1981 .

G. Polya , G . Szego , Problems and Theorems in Analysis I, Spriger -Verlag, Berli n Heidelber g New York , 1978 .

W. W . Rogosinski , Volume and Integral, Olive r an d Boyd , Edinburg h and London ; Interscienc e Publishers , Inc. , Ne w York , 1952.

H. L . Royden , Real Analysis, Th e Macmilla n Company , Ne w York , and Collier-Macmilla n Limited , London , 1968 .

W. Rudin , Principles of Mathematical Analysis, McGraw-Hil l Boo k Company, New York , 1964 .

R. Sikorski , Funkcje rzeczywiste, (Polish) , PWN , Warszawa , 1958 .

P. N . d e Souza , J.-N . Silva , Berkeley Problems in Mathematics, Springer-Verlag, Ne w York , 1998 .

Bibliography - Book s 35 3

[31] E . C . Titchmarsh, The Theory of Functions, Oxfor d Universit y Press , 1939.

[32] E . T . Whittaker , G . N . Watson, A Course of Modern Analysis, Cam -bridge Universit y Press , 1963 .

[33] A . Zygmund, Trigonometric series. Vols. I, II, Cambridg e Universit y Press, 1959 .

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Index

Abel tes t fo r convergenc e o f imprope r integrals, 3 3

absolutely continuou s function , 7 7 almost everywhere , 6 5 approximation theore m fo r measurabl e

functions, 6 5 Arzela theorem , 2 0

Banach indicatrix , 7 8 Banach-Zarecki theorem , 7 8 Bernstein theorem , 8 7 Bessel's inequality , 8 6 Bonnet for m o f secon d mea n valu e the -

orem, 18

Cantor function , 6 6 Caratheodory condition , 5 7 Cauchy theorem , 3 1 Cesaro mea n o f th e Fourie r series , 8 8 change o f variabl e formula , 1 5

for Lebesgu e integrals , 7 9 Chebyshev inequality , 4 4 convergence i n measure , 6 8

Dini-Lipschitz tes t fo r convergenc e o f Fourier series , 8 4

Dirichlet tes t fo r convergenc e o f im -proper integrals , 3 3

Dirichlet-Jordan tes t fo r convergenc e of Fourie r series , 8 4

duplication formula , 4 1

Egorov theorem , 6 7 elementary set , 5 2 equi-integrable functions , 7 4 essential supremum , 7 0

essentially bounde d function , 7 0 Euler 's be t a function , 4 1 Euler 's gamm a function , 3 5

Fatou lemm a fo r Rieman n integrals , 20

Fatou theorem , 9 0 Fejer kernel , 33 4 Fejer theorem , 8 9 Fejer-Lebesgue theorem , 8 9 first mea n valu e theorem , 9 Fourier coefficients , 8 2 Fourier series , 8 2 Frechet theorem , 6 9 Fresnel's integrals , 3 9 function

Lebesgue measurable , 6 4 of bounde d variation , 1 0 of negativ e variation , 1 0 of positiv e variation , 1 0 simple, 6 5 singular, 8 0

Helly selectio n theorem , 1 8 Helly theorem , 1 8 Holder condition , 1 2 Holder inequality , 4 6

integration b y par t s fo r Lebesgu e in -tegrals an d absolutel y continuou s functions, 8 0

Jensen inequality , 4 7 for Lebesgu e integral , 7 6

Lebesgue constants , 8 8

355

356 Index

Lebesgue criterio n fo r Rieman n inte -grability, 6 4

Lebesgue inne r measure , 5 9 Lebesgue point , 8 0 Lebesgue theorem , 6 9 Lipschitz condition , 11 4 Lipschitz conditio n o f orde r en , 1 2 Lipschitz constant , 11 4 Lusin theorem , 6 8

mesh, 3 Minkowski inequality , 4 8 monotone convergenc e theore m fo r th e

Riemann integral , 2 0

Opial inequality , 5 1

pairwise separat e sets , 5 2 Parseval formula , 32 5 part ial integratio n formula , 1 5 point

of approximat e continuity , 8 1 of density , 6 2 of dispersion , 6 2 of oute r density , 8 1 of oute r dispersion , 8 1

Riesz theorem , 6 9

saltus function , 1 4 Schwarz inequality , 4 2 second mea n valu e theorem , 1 8 separate intervals , 5 2 Steffensen inequality , 4 9 step function , 6 Stirling formula , 4 2

total variation , 1 0 t rans la te o f a functio n b y t , 7 6

uniform convergenc e o f a n imprope r integral, 3 5

Vitali set , 6 3 Vitali theorem , 6 9

Wiener theorem , 9 0

Young inequality , 4 5

Titles i n Thi s Serie s

21 W . J . Kaczo r an d M . T . Nowak , Problem s i n mathematica l analysi s III: Integration , 200 3

20 Klau s Hulek , Elementar y algebrai c geometry , 200 3

19 A . She n an d N . K . Vereshchagin , Computabl e functions , 200 3

18 V . V . Yaschenko , Editor , Cryptography : A n introduction , 200 2

17 A . She n an d N . K . Vereshchagin , Basi c se t theory , 200 2

16 Wolfgan g Kiihnel , Differentia l geometry : curve s - surface s - manifolds ,

2002

15 Ger d Fischer , Plan e algebrai c curves , 200 1

14 V . A . Vassiliev , Introductio n t o topology , 200 1

13 Frederic k J . Almgren , Jr. , Plateau' s problem : A n invitatio n t o varifol d

geometry, 200 1

12 W . J . Kaczo r an d M . T . Nowak , Problem s i n mathematica l analysi s

II: Continuit y an d differentiation , 200 1

11 Michae l Mesterton-Gibbons , A n introductio n t o game-theoreti c modelling, 200 0

® 10 Joh n Oprea , Th e mathematic s o f soa p films : Exploration s wit h Mapl e , 2000

9 Davi d E . Blair , Inversio n theor y an d conforma l mapping , 200 0

8 Edwar d B . Burger , Explorin g th e numbe r jungle : A journey int o

diophantine analysis , 200 0

7 Jud y L . Walker , Code s an d curves , 200 0

6 Geral d Tenenbau m an d Miche l Mende s France , Th e prim e number s

and thei r distribution , 200 0

5 Alexande r Mehlmann , Th e game' s afoot ! Gam e theor y i n myt h an d

paradox, 200 0

4 W . J . Kaczo r an d M . T . Nowak , Problem s i n mathematica l analysi s

I: Rea l numbers , sequence s an d series , 200 0

3 Roge r Knobel , A n introductio n t o th e mathematica l theor y o f waves ,

2000

2 Gregor y F . Lawle r an d Leste r N . Coyle , Lecture s o n contemporar y

probability, 199 9

1 Charle s Radin , Mile s o f tiles , 199 9