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होम Problems in mathematical analysis. III

Problems in mathematical analysis. III

W. J. Kaczor, M. T. Nowak

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The best way to penetrate the subtleties of the theory of integration is by solving problems. This book, like its two predecessors, is a wonderful source of interesting and challenging problems. As a resource, it is unequaled. It offers a much richer selection than is found in any current textbook. Moreover, the book includes a complete set of solutions. This is the third volume of ""Problems in Mathematical Analysis"". The topic here is integration for real functions of one real variable.The first chapter is devoted to the Riemann and the Riemann-Stieltjes integrals. Chapter 2 deals with Lebesgue measure and integration. The authors include some famous, and some not so famous, inequalities related to Riemann integration. Many of the problems for Lebesgue integration concern convergence theorems and the interchange of limits and integrals. The book closes with a section on Fourier series, with a concentration on Fourier coefficients of functions from particular classes and on basic theorems for convergence of Fourier series.The book is mainly geared toward students studying the basic principles of analysis. However, given its selection of problems, organization, and level, it would be an ideal choice for tutorial or problem-solving seminars, particularly those geared toward the Putnam exam. It is also suitable for self-study. The presentation of the material is designed to help student comprehension, to encourage them to ask their own questions, and to start research. The collection of problems will also help teachers who wish to incorporate problems into their lectures. The problems are grouped into sections according to the methods of solution. Solutions for the problems are provided. ""Problems in Mathematical Analysis I and II"" are available as Volumes 4 and 12 in the AMS series, ""Student Mathematical Library""

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Student Mathematical Library 021

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STUDENT MATHEMATICAL LIBRARY Volume 21 Problems in Mathematical Analysis III Integration W. J. Kaczor M. T. Nowak t t ( ( ■) ( 1 ( ) 2 o ■ im  Problems in Mathematical Analysis III Integration  This page intentionally left blank  STUDENT MATHEMATICAL LIBRARY Volume 21 Problems in Mathematical Analysis III Integration W. J. Kaczor M. T. Nowak #AMS American Mathematical Society  Editorial Board David Bressoud, Chair Daniel L. GorofT Carl Pomerance 2000 Mathematics Subject Classification. Primary 00A07, 26A42; Secondary 26A45, 26A46, 26D15, 28A12. For additional information and updates on this book, visit www.ams.org/bookpages/stml-21 Library of Congress Cataloging-in-Publication Data Kaczor, W. J. (Wieslawa J.), 1949- [Zadania z analizy matematycznej. English] Problems in mathematical analysis. I. Real numbers, sequences and series / W. J. Kaczor, M. T. Nowak. p. cm. — (Student mathematical library, ISSN 1520-9121 ; v. 4) Includes bibliographical references. ISBN 0-8218-2050-8 (softcover ; alk. paper) 1. Mathematical analysis. I. Nowak, M. T. (Maria T.), 1951- II. Title. III. Series. QA300K32513 2000 515'.076—dc21 99-087039 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904- 2294, USA. Requests can also be made by e-mail to reprint-permission@ams.org. © 2003 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those grante; d to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 08 07 06 05 04 03  Contents Preface vii Part 1. Problems Chapter 1. The Riemann-Stieltjes Integral 3 §1.1. Properties of the Riemann-Stieltjes Integral 3 §1.2. Functions of Bounded Variation 10 §1.3. Further Properties of the Riemann-Stieltjes Integral 15 §1.4. Proper Integrals 21 §1.5. Improper Integrals 28 §1.6. Integral Inequalities 42 §1.7. Jordan Measure 52 Chapter 2. The Lebesgue Integral 59 §2.1. Lebesgue Measure on the Real Line 59 §2.2. Lebesgue Measurable Functions 66 §2.3. Lebesgue Integration 71 §2.4. Absolute Continuity, Differentiation and Integration 79  VI Contents §2.5. Fourier Series 84 Part 2. Solutions Chapter 1. The Riemann-Stieltjes Integral 97 §1.1. Properties of the Riemann-Stieltjes Integral 97 §1.2. Functions of Bounded Variation 114 §1.3. Further Properties of the Riemann-Stieltjes Integral 126 §1.4. Proper Integrals 143 §1.5. Improper Integrals 164 §1.6. Integral Inequalities 207 §1.7. Jordan Measure 228 Chapter 2. The Lebesgue Integral 247 §2.1. Lebesgue Measure on the Real Line 247 §2.2. Lebesgue Measurable Functions 268 §2.3. Lebesgue Integration 281 §2.4. Absolute Continuity, Differentiation and Integration 296 §2.5. Fourier Series 316 Bibliography - Books 351 Index 355  Preface This is a sequel to our books Problems in Mathematical Analysis I, II (Volumes 4 and 12 in the Student Mathematical Library series). The book deals with the Riemann-Stieltjes integral and the Lebesgue integral for real functions of one real variable. The book is organized in a way similar to that of the first two volumes, that is, it is divided into two parts: problems and their solutions. Each section starts with a number of problems that are moderate in difficulty, but some of the problems are actually theorems. Thus it is not a typical problem book, but rather a supplement to undergraduate and graduate textbooks in mathematical analysis. We hope that this book will be of interest to undergraduate students, graduate students, instructors and researches in mathematical analysis and its applications. We also hope that it will be suitable for independent study. The first chapter of the book is devoted to Riemann and Riemann- Stieltjes integrals. In Section 1.1 we consider the Riemann-Stieltjes integral with respect to monotonic functions, and in Section 1.3 we turn to integration with respect to functions of bounded variation. In Section 1.6 we collect famous and not so famous integral inequalities. Among others, one can find Opial's inequality and Steffensen's inequality. We close the chapter with the section entitled "Jordan measure". The Jordan measure, also called content by some authors, vn  Vlll Preface is not a measure in the usual sense because it is not countably additive. However, it is closely connected with the Riemann integral, and we hope that this section will give the student a deeper understanding of the ideas underlying the calculus. Chapter 2 deals with the Lebesgue measure and integration. Section 2.3 presents many problems connected with convergence theorems that permit the interchange of limit and integral; LP spaces on finite intervals are also considered here. In the next section, absolute continuity and the relation between differentiation and integration are discussed. We present a proof of the theorem of Banach and Zarecki which states that a function / is absolutely continuous on a finite interval [a, b] if and only if it is continuous and of bounded variation on [a, 6], and maps sets of measure zero into sets of measure zero. Further, the concept of approximate continuity is introduced. It is worth noting here that there is a certain analogy between two relationships: the relationship between Riemann integrability and continuity, on the one hand, and the relationship between approximate continuity and Lebesgue integrability, on the other hand. Namely, a bounded function on [a, b] is Riemann integrable if and only if it is almost everywhere continuous; and similarly, a bounded function on [a, b] is measurable, and so Lebesgue integrable, if and only if it is almost everywhere approximately continuous. The last section is devoted to the Fourier series. Given the existence of extensive literature on the subject, e.g., the books by A. Zygmund "Trigonometric Series", by N. K. Bari "A Treatise on Trigonometric Series", and by R. E. Edwards "Fourier Series", we found it difficult to decide what material to include in a book which is primarily addressed to undergraduate students. Consequently, we have mainly concentrated on Fourier coefficients of functions from various classes and on basic theorems for convergence of Fourier series. All the notation and definitions used in this volume are standard. One can find them in the textbooks [27] and [28], which also provide the reader with the sufficient theoretical background. However, to avoid ambiguity and to make the book self-contained we start almost every section with an introductory paragraph containing basic definitions and theorems used in the section. Our reference conventions  Preface IX are best explained by the following examples: 1.2.13 or I, 1.2.13 or II, 1.2.13, which denote the number of the problem in this volume, in Volume I or in Volume II, respectively. We also use notation and terminology given in the first two volumes. Many problems have been borrowed freely from problem sections of journals like the American Mathematical Monthly and Mathematics Today (Russian), and from various textbooks and problem books; of those only books are listed in the bibliography. We would like to add that many problems in Section 1.5 come from the book of Ficht- enholz [10] and Section 1.7 is influenced by the book of Rogosinski [26]. Regrettably, it was beyond our scope to trace all the original sources, and we offer our sincere apologies if we have overlooked some contributions. Finally, we would like to thank several people from the Department of Mathematics of Maria Curie-Sklodowska University to whom we are indebted. Special mention should be made of Tadeusz Kuczu- mow and Witold Rzymowski for suggestions of several problems and solutions, and of Stanislaw Prus for his counseling and TeX support. Words of gratitude go to Richard J. Libera, University of Delaware, for his generous help with English and the presentation of the material. We are very grateful to Jadwiga Zygmunt from the Catholic University of Lublin, who has drawn all the figures and helped us with incorporating them into the text. We thank our students who helped us in the long and tedious process of proofreading. Special thanks go to Pawel Sobolewski and Przemyslaw Widelski, who have read the manuscript with much care and thought, and provided many useful suggestions. Without their assistance some errors, not only typographical, could have passed unnoticed. However, we do accept full responsibility for any mistakes or blunders that remain. We would like to take this opportunity to thank the staff at the AMS for their long-lasting cooperation, patience and encouragement. W. J. Kaczor, M. T. Nowak  This page intentionally left blank  Part 1 Problems  This page intentionally left blank  Chapter 1 The Riemann-Stieltjes Integral 1.1. Properties of the Riemann-Stieltjes Integral We start with some basic notations, definitions and theorems. By a partition P of a closed interval [a, b] we mean a finite set of points xo,xi,...,xn such that a = xo < x\ < ... < xn-i < xn = b. The number fi(P) — maxjx^ — %i—i ' i — 1,2,..., 71} is called the mesh of P. For a function a monotonically increasing on [a, b] we write Aa* = a(xi) - a(xi-i). If / is a real function bounded on [a, 6], we define the upper and lower Darboux sums of / with respect to a and relative to P, respectively, by n n U(PJ,a) = ^MiAai, L{PJ,a) = ^m,A^, 2 = 1 2 = 1 where Mi = sup /(x), mi = inf f(x). 3  4 Problems. 1: The Riemann-Stieltjes Integral We also put pb pb / fda = mfU(P,f,a), / fda = sup L(P, f, a), J a J a where the infimum and the supremum are taken over all partitions P of [a, 6], and call them, respectively, the upper and the lower Riemann- Stieltjes integral. If the upper and the lower Riemann-Stieltjes integrals are equal, we denote the common value by J fda and call it the Riemann-Stieltjes integral of / with respect to a over [a, b]. In this case we say that / is integrable with respect to a, in the Riemann sense, and we write / G 11(a). In the special case of a(x) = x we get the Riemann integral. In this case the upper (lower) Darboux sum corresponding to a partition P, and the upper (lower) Riemann integral are denoted, respectively, by U(P,f) (L(P,/)), and J fdx ( Ja fdx j . The Riemann integral of / over [a, b] is denoted by Ja fdx. Moreover, corresponding to every partition P of [a, b] we choose points £i,£2,.-.,£n such that Xi-i < ti < Xi, i = l,2,...,n, and consider the sum n S(P,f,a) = Y,f(U)*CH. 2=1 We say that lim S(P,f,a) = A if, for every e > 0, there is S > 0 such that /jl(P) < S implies that |5(P, /, a) — A\ < e for all admissible choices of fy. In the case when a(x) — x we set n S(PJ) = ^2f(ti)(xi-xi.1). 2=1 Throughout this section, / is always assumed to be bounded and a monotonically increasing on [a, b]. In the solutions we will often use the following theorems (see, e.g., Rudin [28]). Theorem 1. / G 11(a) on [a, b] if and only if for every e > 0 there exists a partition P such that U(P,f,a)-L(P,f,a)<s.  1.1. Properties of the Riemann-Stieltjes Integral 5 Theorem 2. /// is continuous on [a,b], then f G 11(a) on [a, b]. 1.1.1. Suppose a increases on [a, 6], a < c < 6, a is continuous at c, /(c) = 1, and f(x) = 0 for x/c. Show that / G 7£(a) and that £fda = 0. 1.1.2. Suppose / is continuous on [a, 6], a < c < b, a(x) = 0 if x G [a, c), and a(x) = 1 if x G [c, b]. Show that Ja fda = f(c). 1.1.3. Let 0 < a < b and far ifze[a,6]nQ, / (#) = \ [0 ifxG [a,6]\Q. Find the upper and lower Riemann integrals of / over [a, b]. 1.1.4. Let a > 0 and far if ze [-a,a]nQ, \0 ifxG[-a,a]\Q. Find the upper and lower Riemann integrals of / over [—a, a]. 1.1.5. Show that the so-called Riemann function 0 if x is irrational or x — 0, /(x) = ^ 1/g if x = p/g, p G Z, g G N, and p, g are co-prime, is Riemann integrable on every interval [a, b]. 1.1.6. Let / : [0,1] -► R be denned by setting \1 if x= £, nGN, ' 0 otherwise. /(*) Show that JQ f(x)dx = 0. 1.1.7. Show that / : [0,1] -► R defined by lx ~" ix] otherwise is Riemann integrable on [0,1].  6 Problems. 1: The Riemann-Stieltjes Integral 1.1.8. Define JO ifxG[-l,0], Jo ifxG[-l,0), f (x) = < and a(a:) = < \l ifxG(0,l], \l ifxG[0,l]. Show that / G 7£(a) although lim S(P, /, a) does not exist. 1.1.9. Show that if / and a have a common point of discontinuity in [a, 61, then lim S(P, f,a) does not exist. 1.1.10. Prove that if lim S(P, /, a) exists, then / G 11(a) on [a, b] and \\m^S(PJ,a) = J fda. Show also that for every / continuous on [a, 6], the above equality holds. 1.1.11. Show that if / is bounded and a is continuous on [a, 6], then / G 1Z(a) if and only if lim S(P, /, a) exists. M(P)^0 1.1.12. Let \c if a < x < x*, a(x) = < [d if x* < x < 6, where c < d and c < a(a:*) < d. Show that if / is bounded on [a, b] and such that at least one of the functions / or a is continuous from the left at x* and the other is continuous from the right at x*, then / G 11(a) and f(x)da(x) = f(x*)(d-c). I J a 1.1.13. Suppose that / is continuous on [a, b] and a is a step function that is constant on the subintervals (a, ci), (ci, c2),..., (cm, 6), where a < c\ < C2 < • • • < cm < b. Show that /b rn f(x)da(x) = f(a)(a(a+) - a(a)) + £ /(c,)(Q(c+) - Q(c")) fc=i + /(6)(a(6)-a(fc-)).  1.1. Properties of the Riemann-Stieltjes Integral 7 1.1.14. Using Riemann integrals of suitably chosen functions, find the following limits: (a) hm 1 1 1 v } n^oo \n + l n + 2 3n (b) ;im^2f^-^ + ^^ + --- + n^oo \n3 + l3 n3 + 23 n3 + n3 J ' n —>no \ -nA^1 / (d) lim - V(™ + l)(rc + 2) • • • (n + n), n—>oo 72 ( ft ft Tl (e) lim sin — — + sin — — H h sin — n^oo \ n2 + l2 n2 + 22 n2 + ra' / ol/n o2/n on/71 (f) .I"?. ^ + "1T^ + --- + ^ooyn + 1 n + 1/2 n + l/ny' (g) lim yf(\/n)f(2/ri) ... f(n/n), where / is a positive and n—>oo continuous function on [0,1]. 1.1.15. Show that the limit /sin-^-r sin-%- sin -^ \ lim n±l + 5±L + . . . + n+1 1 n^oo y 1 2 n I is positive. 1.1.16. Show that if / is continuously differentiable on [0,1], then n^°° \nt^ W Jo ^\. /e>-/(°>. Using this result, calculate /lfc + 2fc + .-. + nfc 1 \ nm n irn "7 i /e > 0. n-^oo y nfe + i k + 1 J 1.1.17. For k > 0, calculate /'lfc + 3/c + --- + (2n- l)fc hm ( —  8 Problems. 1: The Riemann-Stieltjes Integral 1.1.18. Suppose that / is twice differentiable on [0,1] and f" is bounded and Riemann integrable. Show that j55."'(jf>M*-s|:>(?£i r (i) - r (q) 24 1.1.19. For n G N, define and Show that - l l — n~n + l + n + 2+'"+2n 2 2 2 n_2n + l + 2n + 3+'"' + 4n-l' lim [7n = lim Vn = In 2. n—>oo n—>oo Moreover, using the results stated in 1.1.16 and 1.1.18, show that lim n(ln2 — Un) = - and lim n2(ln2 — Vn) = —. n—»oo 4 n—>-oo 32 1.1.20. Show that if / is Riemann integrable over [a, 6], then / can be changed at a finite number of points without affecting either the integrability of / or the value of its integral. 1.1.21. Show that if / is monotonic and a is continuous on [a, 6], then / G 11(a). 1.1.22. Prove that if / G 11(a) and a is neither continuous from the left nor from the right at a point in [a, 6], then / is continuous at this point. 1.1.23. Let / be Riemann integrable and a continuous on [a, b]. If a is differentiable on [a, b) except for finitely many points and a' is Riemann integrable, then / G 1Z(a) and pb pb / f(x)da(x) = / f(x)a'(x)da J a J a 1.1.24. Let / be Riemann integrable and a be continuous on [a, b] except for finitely many points. If a is differentiable on [a, b] except  1.1. Properties of the Riemann-Stieltjes Integral 9 for finitely many points and a' is Riemann integrable, then / G lZ(a) and / f(x)da(x) = [ f(x)af(x)dx + f(a)(a(a+)-a(a)) J a J a m + J2f(ck)(a(c+) - a(c~)) + f(b)(a(b) - a(b~)), where c^, k = 1, 2,..., m, are points of discontinuity of a in (a, b). 1.1.25. Calculate J_2x2da(x), where a(x) 2' 'x + 2 if -2<z<-l, 2 if - 1 < x < 0, x2 + 3 if 0 < x < 2. 1.1.26. Prove the First Mean Value Theorem. If / is continuous and a is monotonically increasing on [a, 6], then there is c G [a, 6] such that /(x)da(x) = /(c)(a(6)-a(a)). / J a 1.1.27. Show that if / is continuous and a is strictly increasing on [a, 6], then it is possible to choose c G (a, 6) such that the equality in the first mean value theorem (stated above) holds. 1.1.28. (a) Let / be continuous on [0,1]. For positive a and b, find the limit pbe £-+0 + lim / l^ldx. r f± J as X (b) Calculate lim / dx. n-+°° Jo 1 + x 1.1.29. Suppose / is continuous and a is strictly increasing on [a, b\; define F(x)= f f(t)da(t). J a Show that for xG [a, b) y F[x + ft) - F(x) hm —. — r— = fix). h-+o a(x + ft) - a(x)  10 Problems. 1: The Riemann-Stieltjes Integral 1.1.30. Suppose / is continuous on [a, 6], a is both continuous and strictly increasing there, and the limit lm, /(- + *)-/('> . S.{x) h-^o a(x + h) — a(x) da exists and is continuous on [a, b]. Show that 6 df £(x)da(x) = f(b)-f(a). 1.2. Functions of Bounded Variation Recall that the total variation V(/; a, b) of / on [a, b] is y(/;a,6) = sup|^|/(a;i)-/(^1)|l, where the supremum is taken over all partitions P = {a?o,#i, ■■■,xn} of [a, 6]. If V(/; a, 6) < +oo, then / is said to be of bounded variation on [a,b\. We also define Vf(x) = V(f\ a, x), a < x < 6. Clearly, v/(a) = 0 and Vf is monotonically increasing on [a,b\. The following theorem says that a function of bounded variation can be exhibited as a difference of two monotonic functions. Theorem 1. /// is of bounded variation on [a,b], then f(x) -f(a) = p{x) -q(x), where P(x) = 2 M*) + f^ ~ /(fl)) and ?(x) = ^Vf^ ~ fW + f^ are monotonically increasing on [a<b\. The functions p and q are called the positive and negative variation functions of f, respectively. 1.2.1. Show that the function given by „ , fs2cos£ ifxG(0,l], [0 ifx = 0 is differentiable on [0,1] but not of bounded variation.  1.2. Functions of Bounded Variation 11 1.2.2. Show that if / has a bounded derivative on [a, 6], then / is of bounded variation. 1.2.3. Show that the function , x (Vcos^ if xe (0,11, f(x) = { x V J V ; [0 ifx = 0 is of bounded variation on [0,1]. 1.2.4. Show that V(/; a, ft) = /(&)-/(a) if and only if / is monotonically increasing on [a, b\. 1.2.5. For a G R and (3 > 0, define if xe (0,1], Show that / is of bounded variation if and only if a > (3. 1.2.6. Show that if / is of bounded variation on [a, 6], then / is bounded on [a, b]. 1.2.7. If / and g are of bounded variation on [a, 6], then so is their product fg. Moreover, if inf \f(x)\ > 0, then g/f is also of bounded xG[o,6] variation on [a, b]. 1.2.8. Must the composition of two functions of bounded variation be of bounded variation? 1.2.9. If / satisfies a Lipschitz condition and g is of bounded variation, then the composite function / o g is of bounded variation. 1.2.10. Show that if / is of bounded variation on [a, 6], then so is \f\p, l<p< +oo. 1.2.11. Prove that if / is continuous on [a, b] and |/| is of bounded variation on [a, 6], then so is /. Prove also that continuity is an essential hypothesis. 1.2.12. If/ and g are of bounded variation on [a, 6], then so is h(x) = max{f(x),g(x)}.  12 Problems. 1: The Riemann-Stieltjes Integral 1.2.13. We say that / : A —> R, A C R, satisfies a Holder condition (also called the Lipschitz condition of order a) on A if there exist positive constants M and a such that \f(x)-f(x')\<M\x-x'\a for x,x'eA. (a) Show that the function /WJ4) if *e (0,1/2], \o ifx = 0 is of bounded variation on [0,1/2] and does not satisfy a Holder condition. oo (b) Set xn — Yl kin2 k' n — 2, 3,... . Let / be the continuous function k=n on [0, x2) defined as follows: f(0) = f(xn) = 0, f(Xn+2n+1)=^ " = 2,3,..., and / is linear on [xn+1, (xn + xn+i)/2] and [(xn + xn+i)/2,xn]. Prove that / satisfies a Holder condition for every 0 < a < 1, and that / is not of bounded variation on [O,^]- 1.2.14. Suppose that / : [a, oo) —> R is of bounded variation on every interval [a, b], b > a, and put V(/;a,oo)= lim V(f',a,b). b-^oo Show that if V(/;a,oo) < oo, then the finite limit lim f(x) exists. x—>oo Does the opposite implication hold? 1.2.15. For / defined on [a, b] and a partition P = {x0, £'i,... ,xn} of [a, 6], we form the sum vup) = ]T|/(^)-/(^-i)I- Prove that if / is continuous on [a, b], then lim V(/,P) = V(/;a,b), /x(P)->0 that is, for any s > 0 there exists S > 0 such that //(P) < <5 implies V(/;a,&)-V(/,P)<e.  1.2. Functions of Bounded Variation 13 1.2.16. For / defined on [a, b] and a partition P = {xq, x\,. .. ,xn} of [a, 6], we form the sum n W(f,P) = J^(Mi-mi), i=\ where M% = sup f(x), ml = inf /(x). Using the result in the previous problem, show that if / is continuous on [a, b], then lim W(/,P) = V(/;a,&). /*(P)->0 1.2.17. Suppose that / is of bounded variation on [a, 6], with p and g its corresponding positive and negative variation functions as defined in Theorem 1. Suppose also that p\ and q\ are increasing functions on [a, b] such that f = pi — q\. Show that if a < x < y < b, then p(x) -p{y) <Pi(x) -pi(y) and q(x) - q(y) < qi{x) -qi(y). Conclude that V(p; a, b) < V(p\\ a, b) and V(q\ a, b) < V(q\\ a, b). 1.2.18. Suppose that / is of bounded variation on [a, b) and f(x) > m > 0, x G [a,b\. Show that there are two monotonically increasing functions g and h such that /W = 7TT for ^^fa,6]. /z.(x) 1.2.19. Compute the positive and negative variation functions of (a) f(x) = x*-\xl a:G[-l.l], (b) f(x) -cosx, x G [0,2tt], (c) f(x) = x-[xl xe [0,3]. 1.2.20. Assume that / is of bounded variation on [a, b]. Prove that if / is continuous from the right (left) at j~o, then Vf is also continuous from the right (left) at ,tq. 1.2.21. Show that the set of points of discontinuity of a function / of bounded variation on [a J)] is at most countable. Moreover, if  14 Problems. 1: The Riemann-Stieltjes Integral {xn} is the sequence of points of discontinuity of /, then the function g(x) = f(x) — s(x), where s(a) = 0 and s(x) = f(a+) - f(a) + £ (/(*+) " f(x~)) + f(x) - f(x~) Xn<X for a < x < b, is continuous on [a, b]. (The function s is called the s alius function off.) 1.2.22. Let / be of bounded variation on [a, b] and let 1 fx g(a)=0 and g{x) = / f(t)dt, x G (a,b\. x-a Ja Prove that g is of bounded variation on [a, b]. 1.2.23. Show that if / satisfies a Lipschitz condition on [a, 6], then Vf also satisfies the Lipschitz condition with the same Lipschitz constant. 1.2.24. Prove that if / is of bounded variation on [a, b] and enjoys the intermediate value property, then / is continuous. Conclude that if f is of bounded variation on [a, 6], then f is continuous. 1.2.25. Prove that if / is continuously differentiate on [a, 6], then «/(*)= [X\f'(t)\dt- J a 1.2.26. Show that if / is continuous and a monotonically increasing on [a, 6], then the function F(x) = / f(t)da(t), x G [a, 6], ./a is of bounded variation on [a, b\. 1.2.27. If /(x) - lim fn(x) for x G [a, 6], then n—»oo V(/;a,6)< lim V(/„;a,6). n—>oo oo 1.2.28. Suppose that the series Yl an and ]T 6n are absolutely con- 71 = 1 71=1 vergent, and let {xn} be a sequence of distinct points in (0, 1). Prove that the function / defined by /(0) = 0, f(x) =J2an+J2bn, ^ x e (0,1] xn<~.x <x xn<x  1.3. Further Properties ... 15 is continuous at every x ^ xn, n G N, and f(xn) ~ f(x~) = an, f(x+) - f(xn) = bn. Prove also that oo V(f;0,l) = Y,(K\ + K\). n=l 1.3. Further Properties of the Riemann-Stieltjes Integral In this section we consider Riemann-Stieltjes integrals with respect to functions of bounded variation. If a is a function of bounded variation on [a, b], and if a = p — q, where p and q are monotonically increasing, then / f(x)da(x) = f f(x)dp(x) - f f(x)dq(x), provided that / G lZ(p) and / G 1l(q) (see the definition in Section 1.1). This definition does not depend on a decomposition of a into a difference of two increasing functions. Theorem 1. If functions f and a are of bounded variation on [a, b] and one of them is continuous, then f f(x)da(x) = f(b)a(b) - f(a)a(a) - f a(x)df(x). J a J a The above formula is called the partial integration formula. Theorem 2. Suppose f and Lp are continuous on [a, b] and Lp is strictly increasing on [a, b]. If ip is the inverse function of Lp, then / f[x)dx = / /(^(y))#(y). Ja Jif (a) The formula in Theorem 2 is called the change of variable formula. Theorem 3. Suppose that either f is continuous and a is of bounded variation on [a, b], or f and a are of bounded variation on [a, b] and a is continuous. Then / f(x)da(x)\< / \f(x)\dva(x), \J a \ J a  16 Problems. 1: The Riemann-Stieltjes Integral where va(x) denotes the variation of a on [a,x], a < x < b. 1.3.1. Calculate J_ xda(x), where a(x) = < 0 if x = -l, 1 if - 1< x < 2, -1 if 2 < x < 3. 1.3.2. Suppose / is of bounded variation on [0, 2tt] and /(0) = /(27r). Show that each of the integrals r2ir r2-n /»Z7T pZlT / f(x) cos nxdx and / f(x) sin nxdx Jo Jo is not greater than V(f; 0, 27r)/n in absolute value. 1.3.3. Suppose / and g are continuous on [a, 6] and a is of bounded variation, and define P(x) = / f(x)da(x), a<x<b. J a Prove that g(x)d(3(x) = / g(x)f(x)da{x). J a 1.3.4. Let {xn} be a sequence of distinct points in (0,1), suppose oo that cn > 0, Yl cn < °°5 and define 71=1 OO a(x) = YlCnp(x~Xn^ where the function p is given by f 0 if x < 0, [1 if x>0. Prove that if / is continuous on [0, 1], then / /da = Vcn/(x„). ■/o  1.3. Further Properties ... 17 1.3.5. Suppose that a is a continuous function of bounded variation on [a, b] such that for every / continuous on [a, b], rb f(x)da(x) = 0. / J a Show that a is constant on [a, b]. 1.3.6. Let a be monotonically increasing on [0, n] and such that / sinxda(x) = a(7r) — a(0). Show that . Ja(0) if xG[0,7r/2), a{x) = < I a(7r) if x G (7r/2,7r]. 1.3.7. Find a function a monotonically increasing on [0,1] and such that for every / continuous on [0,1]. 1.3.8. Find a function / continuous on [a, b] and such that rb f(x)da(x) = a(b) — a (a) I J a for every a monotonically increasing on [a, b\. 1.3.9. Assume that a is of bounded variation on [a, b) and the functions fn, n = 1,2,..., are Riemann-Stieltjes integrable with respect to a over [a, b]. Prove that if {fn} converges uniformly on [a, b] to /, then / is Riemann-Stieltjes integrable and rb rb / f(x)da(x) = lim / fn(x)da(x). J a n^°°Ja 1.3.10. Calculate lim / nx(l - x2)ndx. n —oc JQ 1.3.11. For a of bounded variation on [0,1], find lim [ xnda{x). n^oc JQ  18 Problems. 1: The Riemann-Stieltjes Integral 1.3.12. Suppose that {an} is a sequence of functions whose total variations are uniformly bounded on [a, b], that is, there is a positive M such that V(an; a,b) < M for all n. Prove that if {an} is pointwise convergent to a on [a, b], then for every / continuous on [a, b], pb pb lim / f(x)dan(x) = / f(x)da(x). n-"°° J a J a 1.3.13. Suppose that {an} is a sequence of functions whose total variations are uniformly bounded on [a, 6], and that {an} is pointwise convergent to a on [a, b]. Suppose also that {fn} is a sequence of continuous functions uniformly convergent on [a, b] to /. Prove that pb pb lim / fn(x)dan(x) = / f(x)da(x). n^°°Ja J a 1.3.14. Prove the following Helly selection theorem. Let {an} be a sequence of functions defined on [a, b] such that |an(a)| < M and V(an\ a,b) < M for n G N. Then {an} contains a subsequence {c*nk} convergent to a function a of bounded variation on [a, 6], and for every continuous function /, pb pb lim / f(x)dank(x) = / f(x)da(x). k^°° J a J a 1.3.15. Prove the following theorem of Helly which generalizes the result in 1.3.12. Let / be continuous and a of bounded variation on [a, b]. If the sequence {an} of functions of uniformly bounded variation converges to a on a set A dense in [a, b] and such that a, b G A, then pb pb lim / f(x)dan(x) = / f(x)da(x). n^°° J a J a 1.3.16. Prove the second mean value theorem. Suppose / is mono- tonic and a is continuous and of bounded variation on [a, b]. Then there is a point c G [a, b] such that I f(x)da(x) = f(a) fCda(x)+f(b) [ da(x) = f(a)(a(c) - a(a)) + f(b)(a(b) - a(c)). 1.3.17. Prove the following Bonnet forms of the second mean value theorem .  1.3. Further Properties ... 19 (a) If / is a positive increasing function on [a, b] and a is a continuous function of bounded variation on [a, 6], then there is a point c <E [a,b] such that f(x)da(x) = f(b) J da(x) = f(b)(a(b) - a(c)). (b) If / is a positive decreasing function on [a, b] and a is a continuous function of bounded variation on [a, 6], then there is a point c £ [a, b] such that r f(x)da(x) = /(a) [ da(x) = f(a)(a(c)-a(a)). dx. 1.3.18. For 0 < a < b, find rb sin(nx) lim / X 1.3.19. For x > 0, prove that (a) if F{x) = Jxx+1 sin(t2)dt, then \F(x)\ < l/x, (b) if F{x) = /*+1 sin(e*)eft, then \F{x)\ < 2/(ex). 1.3.20. Show that if the functions /, a\^a2 are continuous and of bounded variation on [a, b], then 6 />6 pb f(x)d(ai(x)a2(x)) = / f(x)ai(x)da2(x)+ f(x)a2(x)dai{x). J a J a 1.3.21. Show that if / is continuous and of bounded variation on [a, 6], then for a positive integer n, f(x)d((f(x))n) =nf (f(x))ndf(x) J a n + 1 _ ( f(„\\n + \\ T((/W)n+1 - (/(«))n+1) 1.3.22. Suppose that / is continuous on [0,1]. Find the following limits: (a) lim (n fn xnf(x)dx) . n—>oo \ uu " J (b) lim (nfte-nxf(x)dx),  20 Problems. 1: The Riemann-Stieltjes Integral (c) Jo xnf(x)dx lim ^ JQ xnex2dx (d) lirn^ (y/nfl f(x)sm2n(27rx)dx\ , (e) /01/(a:)sin2n(27rx)da lim /0e-2sin2n(2^)d; 1.3.23. Prove the following monotone convergence theorem for the Riemann integral If {/n} is a decreasing sequence of Riemann integrable functions on [a, b] which converges on [a, b] to a Riemann integrable function /, then pb pb lim / fn(x)dx = / f(x)dx. n-+°° J a J a 1.3.24. Prove the following monotone convergence theorem for the lower Riemann integral. If {/n} is a decreasing sequence of bounded functions on [a, b], and if lim fn(x) = 0 for x G [a, 6], then n—>-oo / fn{x)dx ■■ "Ja rb lim / fn(x)dx = 0. n—>-oo 1.3.25. Prove the following Arzela theorem. If {fn} is a sequence of Riemann integrable functions on [a, b] which converges on [a, b] to a Riemann integrable function /, and if there is a constant M > 0 such that \fn(x)\< M for all x G [a, 6] and all n G N, then lim / fn(x)dx = / f(x)dx. J a J a 1.3.26. Prove the following Fatou lemma for Riemann integrals. If {/n} is a sequence of nonnegative Riemann integrable functions on [a, b] which converges on [a, b] to a Riemann integrable function /, then pb pb / f(x)dx < lim / fn(x)dx. J a n-+oo J a  1.4. Proper Integrals 21 1.4. Proper Integrals 1.4.1. For n G N, calculate f4 \x - 1| /"^ /"^ (a) / -: : : .dx, (b) / sii^xdx, / cosn xdx, 7o \x-2\ + \x-3\ J0 J0 (c) J°\lnx\dx, (d) y*- x sin x -5—ax, + cos^ x (e) / t^n2nxdx, (f) / —^—:dx 0 7o sin x + cos x (g) / „:-n_ , ^.^ sinn x + cosn x 1.4.2. For n G N, use the integral JQ (l — x2) dx to calculate 1 /n\ 1 /n\ 1 /n\ , .„ 1 •■ + (-1) \z 1 VOy 3 \lj 5\2J v y 2n+l \n 1.4.3. Suppose that a function / has an indefinite integral (or anti- derivative) on an interval I; that is, there is a different iable function F such that F'(x) = f(x) for x G I. Show that if a one-sided limit of / at xq G I exists and is equal to a, then f(xo) = a. 1.4.4. Let / be defined by t( \ fsinx if x ^°' [c it i = 0, where c £ [—1,1]. For which values of c does there exist an antideriv- ative of /? 1.4.5. Let xn = l/y/n for n G N. Construct a function / continuous on (0,1] and such that / > 0 on [x2fc, ^2/c-i], / < 0 on [x2k+i,x2k] and F(x2fc-i) - ^(^2/c) = F(x2k+i) ~ F(x2k) = 1/fc, where F is an antiderivative of /. Extend the function / to [0,1] by setting /(0) = 0. Prove that / has an antiderivative on [0, l], but |/| does not. 1.4.6. Suppose / is continuous on [0,1]. Show that r 7r r \ xf(sinx)dx=— / f(sinx)dx. Jo 2 J0  22 Problems. 1: The Riemann-Stieltjes Integral Using this equality, compute I • 2r? xsm x dx, n G N. o sin2n x + cos2n x 1.4.7. Assume that / is continuous on [—a, a], a > 0. Show that (a) /a pa f(x)dx = 2 / f(x)dx, if / is even, -a Jo (b) f(x)dx = 0, if / is odd. / J —c f J a 1.4.8. Let / : R —» R be continuous and periodic with period T > 0. Prove that for every real a, a+T /.T f(x)dx = / f(x)dx. Jo 1.4.9. Let / : R —» R be continuous and periodic with period T > 0. Prove that for every a < b, lim / f(nx)dx = —— / f(x)dx. n^°° J a 1 Jo 1.4.10. Suppose that / G C([-l, 1]). Find the following limits: (a) J^£/0n/(sinx)cfe, (b) lim ±f£ f(\smx\)dx, (c) lim fn x/(sin(27rnx))dx. 71—KX) U 1.4.11. For / G C([a,6]), find the following limits: (a) lim f /(a:) cos(nx)dx, lim f /(a:) sin(nx)<ia:, n—KX) n—>oo (b) lim f f(x) sin2(nx)dx. n—>oo a 1.4.12. Suppose / G C([0,oo)) and set an— I f(n + x)dx for n = 0,1,.... Suppose also that lim an = a. Find the limit lim L f(nx)dx.  1.4. Proper Integrals 23 1.4.13. For a function / positive and continuous on [0,1], compute f1 m dx Jo f(x) + f(l-x) 1.4.14. Show that, if / is continuous and even on [—a, a], a > 0, then 1.4.15. Show that if / is nonnegative and continuous on [a, b] and rb f(x)dx = 0, / J a then / is identically zero on [a, b]. 1.4.16. Show that if / is continuous on [a, b] and for each a,/3, a < a<f3<b, / f(x)dx = 0, J a then / is identically zero on [a, b]. 1.4.17. Let / be continuous on [a, b] and such that / f(x)g(x)dx = 0 J a for every function g continuous on [a, b]. Show that / is identically zero on [a, b]. 1.4.18. Let / be continuous on [a, b] and let rb (x)g(x)dx = 0 / /(a J a for every function g continuous on [a, b] and such that g(a) = g(b) = 0. Show that / is identically zero on [a, b]. 1.4.19. Suppose that / is continuous on R. Show that (a) if f* f(t)dt = 0 for every x G R, then / is an odd function, X (b) if f* f(t)dt = 2 J f(t)dt for every xGl, then / is an even o function, (c) given T > 0, if j*+T f(t)dt = /QT f(t)dt for every x G R, then / is periodic with period T > 0.  24 Problems. 1: The Riemann-Stieltjes Integral 1.4.20. Compute rn+l xdx (a) lim (n4 f n-+°° V Jn Jn X5 + 1 2n xdx (b) lim .... 71-+00 y Jn Xb + ly (c) lim / In I x H ) dx, r27T xdx (d) lim -I n-»-oo ^3^ 7^ arctan(nx) 1.4.21. Find the following limits: (a) lim / e-Rsintdt, (b) lim / yfx sin xdx, n-+°° Jo (c) lim / dx. n-+°° Jo v 1 + # 1.4.22. For a function / continuous on [0,1], find lim / f(xn)dx. n-+°° Jo 1.4.23. Show that, if / is Riemann integrable on [a, 6], then there is 0 e [a,b] such that / f(t)dt = f f(t)dt. Ja JO 1.4.24. Let / be continuous on [a, b] and let / f(x)dx = 0. J a Show that there is 0 £ (a, b) such that / f(x)dx = f(0). J a 1.4.25. Let / G C([a, 6]), a > 0, and let / f{x)dx = 0. J a  1.4. Proper Integrals 25 Show that there is 0 G (a, b) such that f f(x)dx = 0f(B). J a 1.4.26. Suppose /, g G C([a,b]). Show that there is 9 G (a, b) such that 0(0) / f(x)dx = /(0) / <?(z)<fc. «/a «/a 1.4.27. Suppose /,pE C([a, &]). Show that there is 0 G (a, 6) such that g{9) [ f(x)dx = f{9) f g(x)dx. Ja JO 1.4.28. Suppose / and g are positive and continuous on [a, b]. Show that there is 9 G (a, b) such that Ja /W^ Jfl p(^)dx 1.4.29. Let / be positive and continuous on [0,1]. Prove that for every n G N there is 9(n) such that I r1 re(n) r1 — / f(x)dx = / f(x)dx + / f(x)dx. n J0 J0 Jl-6{n) Find the limit lim (n9(n)). n-+oo 1.4.30. Let / G ^([O,1]). Show that there is a 0 G (0,1) such that f f(x)dx = f(0)+1-ff(6). 1.4.31. Let / G C2([0,1]). Show that there is a 0 G (0,1) such that jf^(x)dx = /(0) + i//(0) + i///(6l). 1.4.32. Suppose / G CHiO, 1]) and f (0) ^ 0. For x G (0,1], let 9(x) be such that r f(t)dt=f(6(x))x. Jo Find the limit x-+0+ X  26 Problems. 1: The Riemann-Stieltjes Integral 1.4.33. Suppose / is continuous, nonnegative and strictly increasing on [a, b]. For p > 0, let 0(p) denote the unique number such that (f(o(P)r = ^ j\f(x)Ydx. Find lim 6(p). p-+oo 1.4.34. Suppose / is continuous on [a, b] and such that / xnf(x)dx = 0 Ja for n = 0,1, Prove that / is identically zero on [a, b]. 1.4.35. Suppose / is continuous on [a, b] and such that xnf(x)dx = 0 / J a for n = 0,1,..., N. Prove that / has at least N + 1 zeros in [a, b]. 1.4.36. Suppose that / G C([—a, a]), a > 0. Show that (a) if /•a x2nf(x)dx = 0 for n = 0,l,..., then / is odd on [—a, a], (b) if I x2n+1f(x)dx = 0 for n = 0,l,..., then / is even on [—a, a]. 1.4.37. For / continuous on R, find limi / (f(x + h)-f(x))dx. h-+o h Ja 1.4.38. For / continuous on R and a < 6, define g(x) = [ f{x + t)dt. Ja Find the derivative of g. 1.4.39. Find the following limits:  1.4. Proper Integrals 27 (a) lim — / In ( 1 + -^ ) dt, 1 /INT f C0S^ 7 (b) lim x / —^rdt, t2 , x fnx sin\/^^ c lim Jo ^_ / 1 fX P(t) \ (d) lim I — / In dt J , where a > 1, and P and Q are polynomials positive on R+. 1.4.40. Find the following limits: ( i r i \ (a) lim / dt , v} - "\inx7o ^rr^ y x—>-oo (b) lim (- r (1 + sin t)*dt (c) lim (\ J* tl+tdt), *-0+ \X2 Jo J (d) lim ( / ef2dt ) l/(*2 1.4.41. Show that if / is continuous on [0,1], then lim (f \f(x)\*dx) = max |/(x)|. p-^°° \./o / *e[o,i] 1.4.42. Suppose that a real-valued function f(x,y) is continuous on a rectangle R = [a, 6] x [c, d]. Show that i(y) = / f(x,y)dx J a is continuous on [c, d]. 1.4.43. Suppose that a real-valued function /(#, y) defined on a rectangle R = [a, b] x [c, d] is Riemann integrable over [a, 6] for each y G [c, d], and the partial derivative -?f- is continuous on R. Prove that ±Jaf(X,y)dx = Jad/y(X,y)dx.  28 Problems. 1: The Riemann-Stieltjes Integral 1.4.44. Let / be positive and continuous on [0,1]. Find i/p hm Qf (f(x)Ydx 1.4.45. Let / be positive and continuous on [0,1]. Find lim ( J (f(x)rdx) . 1.4.46. Prove that for every positive integer N the equation 1! + 2! +'"+ (2AT)!/ has a solution in the interval (iV, 2N). I e~*U + ^ + ^7 + ' ' ' + T^TTTT )dt = N 1.4.47. Let P be a polynomial of degree n such that I Jo xkP(x)dx = 0 for fc = l,2...,n. /o Prove that / {P{x))2dx={n + l)2(j P{x)dx\ . 1.4.48. Show that if / is continuous on R = [a, b] x [c, d], then / I / f(x,y)dx\dy = I / f(x,y)dy J dx. 1.4.49. Prove that for 0 < a < 6, i 1 xb - xa , , 1 + 6 dx = In - /o In x 1 + a 1.5. Improper Integrals Assume that / is defined on [a, oo) and is Riemann integrable over any finite interval [a, b]. Then we define N ix, /•oo /*o / f(x)dx = lim / f{x)dx J a b^°° J a provided that this limit exists and is finite. In this case we say that the improper integral on the left converges; otherwise, we say that it  1.5. Improper Integrals 29 diverges. The improper integral J_ f(x)dx is defined analogously. The integral J_ f(x)dx is defined by /+oo pa /* + oo f(x)dx = / f(x)dx + / f(x)dx -oo J—oo Ja provided that both improper integrals on the right converge. The definition does not depend on the choice of a. For / defined on [a, b) and Riemann integrable over each closed subinterval of [a, 6), the improper integral J f(x)dx is defined as rb rb—ri po fo—ri / f(x)dx = lim / /(; x)dx, provided the limit is finite. If / is defined on (a, b] and Riemann integrable over each closed subinterval of (a, 6], the improper integral Ja f(x)dx is defined in a similar way. 1.5.1. For n G N and positive a, calculate /•2tt (a) / (b) / ln(sinx)dx, sin x + cos4 x Jo (c) / xn(l-x)adx, a>-l, (d) / (-lnx)ndx, JO </0 (e) 7o f Jo 1 - (1 - x/n)n dx dx, /o (1 + ^2)™' (1) L ^T^dx> (k) / ln(l + cosx)dx, Jo 1.5.2. For 0 < a < 1, define fa{x) = (f) / xnlnn (h) / IH^I xdx, dx lnx -dx, 0) /„ (^ + a2)2 2 * rai — LxJ — a [11 — X 0<x < 1.  30 Problems. 1: The Riemann-Stieltjes Integral Show that / Jo fa(x)dx = a In a. 1.5.3. Suppose / is monotone on the interval (0,1) and the improper integral J0 f(x)dx exists. Show that lim m±m±^±iSr^i = [ f(x)d* Jo 1.5.4. Suppose / is monotone on the interval (0,1) and one of the limits lim f(x) or lim f(x) is finite. Show that if x^0+ x^l- lim m±m±^i±nm. exists as a finite limit, then the improper integral J0 f(x)dx exists. 1.5.5. Show by example that in the preceding problem the assumption that one of the one-sided limits is finite cannot be omitted. 1.5.6. Using the result in 1.5.3, find (a) lim 71—KX) fl /U\ r n, . 7T . 2tt . (n- 1)tt (b) lim \ sin — sm—...sin , \ k=l \ k=l / 1.5.7. Suppose that the function / : (0,1] —> R is monotone and for some a G R the improper integral JQ xaf(x)dx exists. Show that lim xa+lf (x) = 0. 1.5.8. Verify whether the following improper integrals converge or diverge:  1.5. Improper Integrals 31 (^ r^-, (b) r^dx, J 2 x In x J0 1 + xz f1 f1 dx (c) J^-]nx)adx, aeR, (d) ^ xa{_lnx)b> a>b e (e) / r xdx + £ sin x 1.5.9. Suppose that / and g are positive on [a, oo) and J g(x)dx diverges. Show that at least one of the integrals / f(x)g(x)dx, / -rr\dx Ja Ja f{X) diverges. 1.5.10. Prove the following Cauchy theorem. In order that the improper integral Ja°° f(x)dx converge, a necessary and sufficient condition is that, given e > 0, there is a^ > a such that for ci2 > cl\ > ao, ra2 I f(x)dx\ ./ai < £. 1.5.11. Show that the improper integral J f{x)dx converges if and only if for every increasing sequence {an}, an > a, diverging to infinity the series (1) 2_. / f{x)dx, where ao = a, n=1Jan-1 converges. Moreover, in the case of convergence, /•oo °° pan / f(x)dx = J2 / f(z)dx. Show also that if / is nonnegative, a sufficient condition for the convergence of the improper integral is that there is an increasing sequence {an}, an > a, diverging to infinity for which the series (1) converges. 1.5.12. For positive a, study the convergence of the integral dx 1 + xa sin2 x f Jo  32 Problems. 1: The Riemann-Stieltjes Integral 1.5.13. Suppose / is positive on [0, oo) and J0 f(x)dx exists. Must f(x) tend to zero as x —» oo? 1.5.14. Suppose / is positive, differentiable on [a, oo), and |/'(a:)| < 2 for x > a. Does the convergence of J f(x)dx imply that f(x) tends to zero as x —» oo? 1.5.15. Prove that if / is uniformly continuous on [a, oo) and the improper integral f f{x)dx converges, then lim f(x) = 0. a x—>oo 1.5.16. Assume that / : [0, oo) —» [0,oo) is monotone decreasing. Prove that if / Jo f(x)dx < oo, /o then lim xf(x) = 0. Show by example that the converse does not X—KX) hold; that is, the condition lim xf(x) = 0 does not imply the con- x—»oo vergence of J0 f{x)dx. 1.5.17. Assume that / : [1, oo) —> (e, oo) is monotone increasing and f°° dx /NT. 1 1 f00 dX (a) Prove that also / —■——■— = oo. J1 xlnf(x) (b) Give an example of a function / satisfying the above assump- dx :\nf(xW(\nf(x)) 1.5.18. Let / be a continuous function on [0, oo) such that lim (7(x)+ f f{t)dt x^°° \ Jo j exists as a finite limit. Prove that lim f(x) = 0. X—KX) 1.5.19. Let / be a nonnegative and continuous function on [0,oo) and /•oo tion for which / —-— ^ — converges. f(x)dx < oo. Jo Prove that f Jo 1 fn lim — / xf(x)dx = 0. twoo n 70  1.5. Improper Integrals 33 1.5.20. Suppose that / is uniformly continuous on [a, oo) and the integrals J f(t)dt are uniformly bounded; that is, there is M > 0 such that J a f(t)dt < M for x G [a,oo). Show that / is bounded on [a, oo). 1.5.21. Prove that if J^°(f(x))2dx and f^°(f"(x))2dx converge, then f^° (f'(x))2dx also converges. 1.5.22. Prove the following Abel test for convergence of improper integrals. Assume that the functions / and g defined on [a, oo) satisfy the following conditions: (1) the improper integral J f(x)dx exists, (2) g is monotone and bounded on [a,oo), Then the improper integral J a f(x)g(x)dx converges. 1.5.23. Prove the following Dirichlet test for convergence of improper integrals. Assume that the functions / and g defined on [a, oo) satisfy the following conditions: (1) / is properly integrable on each interval [a, 6], b > a, and the integrals J f(x)dx are uniformly bounded, that is, there exists C > 0 such that J a < C for all b > a. f(x)dx\ (2) g is monotone and lim g(x) = 0. Then the improper integral /•OO / f(x)g(x)dx J a converges.  34 Problems. 1: The Riemann-Stieltjes Integral 1.5.24. For a > 0, study the convergence of the following integrals: M f^. (b, f ^ — dx, (c) r sm(x2)dx, (d) ^ es'mx^^dx, /OO sinx 1.5.25. Assume that / : [a, oo) —^ IR is continuous and periodic with period T > 0, and g : [a, oo) —^ IR is monotonic and lim g(x) = 0. x—»oo Prove that if J f(x)dx = 0, then the integral J f(x)g(x)dx converges. Moreover, prove that if J^ f(x)dx ^ 0, then the improper integral J f(x)g(x)dx converges if and only if J g(x)dx converges. 1.5.26. Use the result in the previous problem to study the convergence of the following integrals: (a) / ^ol"J''ea»xdx, f Jo (b) / Jo sin(sinx) x si o x x sin(sinx)cSina.^ 1.5.27. For a > 0, study the convergence of the integral f°° sinx , / dx. J0 xa + sin x 1.5.28. Show that if J xf(x)dx, a > 0, exists, then also J f(x)dx exists. 1.5.29. Assume that / is monotone on [0, oo) and the improper in- "OC 0 tegral J0°° f(x)dx exists. Show that oo „c lim h V f(nh) = \ *-o+ ^n Jo f(x)dx. Use this result to find OO 7 A ^ 1 + /i2n2 ' n = l  1.5. Improper Integrals 35 1.5.30. For a > 0, set T(a) ~ldx. Show that r(a) is finite for all positive a. (T(a) is called Euler's gamma function). 1.5.31. Use the result in 1.5.29 to show that for a > 0, Via) = hm — -t —. rwoo a(a + 1) ... (a + n — 1) 1.5.32. Use the formula given in the previous problem to show that for a > 0, r(0).^n^(l+|)-, 71=1 where 7 is Euler's constant (see, e.g., I, 2.1.41). 1.5.33. Show that 1.5.34. Prove that f Jo r Jo S1I1X 7T dx = —. x 2 dx = 1.5.35. Let f(x, y) be a function defined on [a, 00) x A, where AcR. We say that the integral Ja°° f(x,y)dx converges uniformly on A if, given e > 0, there is ao > a such that / f(x,y)dx- / f(x,y)dx\ \J a Ja < e for all b > ao and y G A. Show that if there is a function <p(x) such that \f(x,y)\<ip(x) for x G [a, 00), y G A and j°° Lp(x)dx converges, then the improper integral J^° f(x,y)dx converges uniformly on A. 1.5.36. Prove the following test for uniform convergence of an improper integral. Suppose that Ja°° f(x,y)dx converges uniformly on  36 Problems. 1: The Riemann-Stieltjes Integral A and g(x,y) is monotonic with respect to x and is bounded on [a, oo) x A. Then J a f(x,y)g(x,y)dx converges uniformly on A. 1.5.37. Conclude the following test from the previous problem. If the improper integral J^° f(x)dx converges and g(x,y) is monotonic with respect to x and is bounded on [a, oo) x A, then /•OO / f{x)g(x,y)dx J a converges uniformly on A. 1.5.38. Suppose that there is a positive C such that / f(x,y)da J a < C for all b > a, y G A, and g(x,y) is monotonic with respect to x and g(x,y) converges to zero as x —* oo uniformly on A. Then f J a f(x,y)g(x,y)dx converges uniformly on A. 1.5.39. Study the uniform convergence of the following improper integrals: . f°° sin ax (a) / dx, a G[a0, oo), a0 > 0, Jo x .. N f°° sin ax (b) / dx, ae[0,oo), Jo x cosaix2 + 1) 7 ^ v 'dx, aeR, (c) (d) Jo f Jo x2 + l e ax cosx2dx, a G (0,oo). 1.5.40. Assume that the improper integral J^° f(x,y)dx converges uniformly on A. Prove that if f(x,y) converges to <p(x) as y —>  1.5. Improper Integrals 37 2/0 uniformly on every interval [a, 6], then the integral J ip(x)dx converges and /•OO /«00 lim / f(x,y)dx= / (p(x)dx. y^yo J a J a 1.5.41. Consider the example < ( \ \^e~^ for x >0' [0 lor x = 0, to show that the assumption of the uniform convergence of the improper integral J f(x,y)dx cannot be dropped from the theorem of 1.5.40. 1.5.42. Show that x 2' , Z"00 sinx 7r (a) hm / e yxdx ■■ y^o+Jo f°° 7T f°° (b) lim / sinx2 arctan(yx)dx = — / sinx2dx, y^oo 70 2 70 , N ,. Z"00 sinx / x\-n Z"00 sinx _ (c) hm / 1 + - ) dx = e xdx, n^ooj0 X V TlJ J0 X (a\ v f°° arctan(yx) tt (d) lim / 2 /I Tdx=o' /•OO (e) lim / y2sinxe~y2x2dx = 0. 1.5.43. Suppose that a real-valued function f(x,y) is continuous on [a, oo) x [c, d] and the improper integral Ja°° f(x,y)dx converges uniformly on [c, d]. Then the function /•OO ^"(y) = / f(x,y)dx J a is continuous on [c, d]. 1.5.44. Suppose that a real-valued function f(x,y) defined on R = [a, oo) x [c, d] is such that Ja°° f(x,y)dx converges for every y G [c,d] and -^- is continuous on R. Suppose also that the integral Ja°° ~d~{xiV)d% converges uniformly on [c,d]. Prove that d_ dy , / f(x,y)dx = / — (x,y)dx.  38 Problems. 1: The Riemann-Stieltjes Integral 1.5.45. Use the result in 1.5.44 to calculate /•OO (a) / e-ax2x2ndx, a > 0, n € N, Jo dx a > 0, n G N, /•oo (b) A (^+x2)«+l' , Z"00 sin(ax) _ (c) / —-—-e xdx, a G R, Jo ri_ Jo . . ^ cos(ax) _x (d) / —-e xdx, a G R. 1.5.46. Prove that /OO e~x2/2 cos(yx)dx = V2^e~y2/2, y G R, -oo (b) fe-^4 = ^-2^ y>0. Jo V^ 1.5.47. Show that if a real-valued function f(x,y) is continuous on [a, oo) x [c, d] and the improper integral /•OO / f(x,y)dx J a converges uniformly on [c, d], then / (/ f(x^y)dx)dy= f / /(z,y)dyjcte. 1.5.48. Use the above theorem to find /•oo p — bx p — ax (a) / dx, a,b>0, Jo z n . cos(6x) - cos(ax) (b) / —'—^ —-dx, 0 < a < b. f Jo 1.5.49. Assume that / G C1([0,oo)), /' is monotonic on [0, oo) and lim f(x) = I exists as a finite limit. Show that for a, b > 0, Jo /(fa)-/(oi)-ds = (Z-/(0))ln^. x a 1.5.50. Let f(x, y) be continuous on [a, oo) x [c, oo) and suppose that (1) the improper integral J°° f(x,y)dx converges uniformly on each interval [c, d],  1.5. Improper Integrals 39 (2) the improper integral J f(x,y)dy converges uniformly on each interval [a, 6], (3) the improper integrals Ja°° \f{x,y)\dx and Jc°° \f(x,y)\dy converge for y > c and x > a, respectively, (4) at least one of the improper integrals U \f(x,y)\dy\dx, J (J \f(x,y)\dx\dy converges. Then /oo / /«oo \ /«oo / /«oo \ (/ f{x,y)dy)dx= ( f(x,y)dx\dy, and both integrals converge. 1.5.51. Prove that f°° 2 , 1 /"^ COSX , 1 R I cosx dx = - / —— d:r = -\ — Jo 2 Jo V* 2V2 and f° . 2j 1 /""sina:, 1 fiF / sins da; = - / ——cte = -\ —• Jo 2Jo yfi 2V2 (The improper integrals J0 cos x2dx and J0 cos x2dx are called Fres- nel's integrals.) 1.5.52. Suppose that f(x,y) is defined on [a, b) x A and for each y G A, / is Riemann integrable over each interval [a, 6 — 77], where 0 < 77 < 6 — a. Suppose also that f(x,y) converges to <p(x) as y —> 2/0 uniformly on each of these intervals. If the improper integral J f(x,y)dx converges uniformly on A (that is, given e > 0, there is 770 such that / f(x,y)dx Jb-T] < 6 for all 0 < r] < ?7o < b — a and y G A) then rb pb lim / f(x,y)dx = / (f(x)dx. y^yo J a J a  40 Problems. 1: The Riemann-Stieltjes Integral 1.5.53. Suppose that fn : [a, b) —» R, n G N, is Riemann integrable on each interval [a, b — rj], where 0 < 77 < b — a, and /n(x) converges to <p(x) as n —> 00 uniformly on each of these intervals. Suppose also that there is a positive function f(x) such that |/n(a:)| < f(x) for all x G [a, 6) and n G N, and the integral J f(x)dx exists. Then 1.5.55. Show that pb rb lim / fn(x)dx = / (f(x)dx. J a J a /•oo lim / e~x ^°°Jo T(x) = lim / n^°° Jo 1.5.54. Show that lim / e~x° dx = 1. 1 - - ) tx~ldt, x > 0. n 1.5.56. Prove that 2 rm dx — —. 2 1.5.57. Show that for 0 < a < 1, / -—dx = - + y -ir —r + v a + /c a — k I sin 7ra 1.5.58. Show that for a, 6 G (0,1), poo ~a —1 _ ~.6—1 / dx — 7r(cot7ra — cot nb). Jo 1~x 1.5.59. Express the integrand as a power series to find (a) r^iz^, (b) r^ii^, Jo x Jo x (c) [lh^±^ldx, (d) f11**^1-*^. Jo x Jo x 1.5.60. Use the identity —-/'■ i + l Jo 2n , , , x2ndx, n = 0,1,2,..., to find the sum of the series  1.5. Improper Integrals 41 /I 1 71 = 0 X <b>J + E (*rnn *^t 71=1 X 1.5.61. Use the result in 1.5.1(c) to determine the sum of the series ~ 1 wE (2n + l)(2n + 2)(2n + 3); (b)E 71=0 ^2n(2n+l)(2n + 2)- 1.5.62. Show that -1 oo f°° xe~x 1.5.63. Find the value of the integral / —dx. Jo l + e x 1.5.64. For positive a and 6, define B{a,b) = [ xa-l(l-x)b-ldx. Jo (B(a,b) is called Euler's beta function.) Prove that for 0 < a < 1, 1.5.65. Show that for a, b > 0, B(„ h\ - T{a)m and conclude that T(a)T(l-a) = —^— sin7ra for 0 < a < 1. 1.5.66. Find lim aT(a). a-+0+ 1.5.67. Calculate J1 In T(a)da. dy= -—  42 Problems. 1: The Riemann-Stieltjes Integral 1.5.68. For a > 0, derive the following duplication formula: o2ar(a)r(a+l/2) 2^. (a) / t&na xdx = JZ~7^\^ H < ^ Jo (b) r ^— = ^(r(i/4))2, 7o T(2a) 1.5.69. Verify the following equalities: ' ' 7T 2cos(7ra/2)' da: 1 \/3 — cos x 4v/7r ,tt/2 (c) / sin0"1 xdx = 2a~2£(a/2, a/2), a > 0. Jo 1.5.70. Derive the following Stirling formula: lim . ; = 1. n~^°° V27rnn+1/2e-n 1.5.71. Show that for a > 0, (a) Jo V T(a) J0 V {l + x)aJ x ' 1.5.72. Find the following limits: v^r(x + i/2) (a) lim r^.n ' (b) lim xaB(a,x), a > 0. a;—>-oo 1.6. Integral Inequalities 1.6.1. Prove the following Schwarz inequality. If / and g are Rie- mann integrable on [a, 6], then If f(x)g(x)dx\ < J f(x)dx J g\x)dx. Moreover, if / and g are continuous on [a, &], then the equality holds if and only if there are Ai and A2 such that |Ai| + |A21 > 0 and Xif(x) = X2g(x) for x £ [a, b].  1.6. Integral Inequalities 43 1.6.2. Show that if / is Riemann integrable on [a, 6], then I / f(x)sinxdx\ +1 / f(x)cosxdx\ < (b — a) / f2(x)dx. 1.6.3. Show that if / is positive and Riemann integrable on [a, 6], then rb rb {b-«^L!{x)d*Lm Moreover, if 0 < m < f(x) < M, then pb rb / f(x)dx J a J a dx (m + M) 2 < ,„,/ (*>-") 2 f(x) ~~ 4mM 1.6,4. Let / and g be Riemann integrable on [a, b] and let mi < f(x) < Mi and 777,2 < g(x) < M2, x G [a, b]. Show that / f(x)g(x)dx - — - / f(x)dx / g(x)dx ~a J a \P- a)1 Ja Ja < -(Mi -mi)(M2 -m2). 1.6.5. Suppose / G C1^]), /(a) = f(b) = 0 and f* f2(x)dx = 1. Show that and j< / (f'(x))2dx I x2f(x)dx. 1.6.6. Find min / (l+x2)f2(x)dx, f£Aj0 where A=lfeC([0,l]):J f(x)dx = l\ and find a function for which the minimum is attained.  44 Problems. 1: The Riemann-Stieltjes Integral 1.6.7. Assume that / : [a, b] —► [ra, M] and Ja f(x)dx = 0. Prove that f2(x)dx< -mM(b-a). I Ja 1.6.8. Show that if / is continuous on [0,1], differentiate on (0,1), /(0) = 0, and 0 < f'(x) < 1 on (0,1), then ' / f(x)dx) > I (f(x))3dx. kJo J Jo Show also that equality occurs if and only if f(x) = x. 1.6.9. Let / G C([a,b]) be monotonically increasing on [a, b]. Show that for x G (a, 6), ^L_ £ mdt < _1_ ^6 /(t)d4 < _1_ £ fm. 1.6.10. Prove the following Chebyshev inequality: If functions / and g are either both increasing or both decreasing on [a, 6], then rb -i rb ^ rb b I rb I [b 1 fb / f(x)dx- / g{x)dx < / f(x)g(x)dx. -a Ja b-a Ja o-aJa If one of the functions is increasing and the other decreasing, then the above inequality reverses. 1.6.11. Prove the following generalization of the Chebyshev inequality: Let p be positive and Riemann integrable on [a, b]. If the functions / and g are either both increasing or both decreasing on that interval, then rb nb pb rb / p(x)f(x)dx / p(x)g(x)dx < / p(x)dx / p(x)f(x)g(x)dx. Ja Ja Ja Ja If one of the functions is increasing and the other decreasing, then the above inequality reverses. 1.6.12. Assume that / and g are monotonically increasing on [0, a]. Show that pa pa / f(x)g(x)dx > /(a - x)g(x)dx. Jo Jo  1.6. Integral Inequalities 45 1.6.13. Let q be positive and decreasing on [a, 6], a > 0. Show that f*xq2(x)dx < f*q2(x)dx f*xq(x)dx ~ jbaq(x)dx 1.6.14. Show that if / is a convex function on [a, 6], then /(^)(^«></>^<^>±M<>-«). 1.6.15. Given positive real numbers x and y, let A, G and L be their arithmetic, geometric and logarithmic means, respectively (see, e.g., II, 2.5.41 for the definition of the logarithmic mean). Use the previous result to show that for x ^ y, AL < GA if both x and y are at least e3/2 and AL > GA if both x and y are at most e3/2. 1.6.16. Show that if / G C([a, 6]) is positive and strictly concave on [a, 6], then / 6 1 f(x)dx > -(b - a) max /(a). Z xG[a,6] 1.6.17. Suppose /, g are continuously differentiate on [0,6], /', g' are nonnegative on [0,6], and / is nonconstant with /(0) = 0. Then for 0 < a < 6, f(a)g(b)< [ag(x)f\x)dx+ f g\x)f(x)dx. Jo Jo The equality holds if and only if a = 6 or g is a constant function. 1.6.18. Suppose / G C1([0, c]), c > 0, is strictly increasing on [0, c] and /(0) = 0. Show that for x G [0,c], / /(t)dt+ / f-\t)dt = xf(x), Jo Jo where f~l denotes the inverse of /. 1.6.19. Use the result in the previous problem to prove the Young inequality. Under the assumption of 1.6.18, rb pa pb / /(t)dt+ / f~\t)dt>ab Jo Jo for any a G [0, c] and 6 G [0, /(c)]. Moreover, the equality holds if and only if 6= /(a).  46 Problems. 1: The Riemann-Stieltjes Integral 1.6.20. Show that for a, 6 > 0, (1 + a) ln(l + a) - (1 + a) + (eb - b) > ab. 1.6.21. Prove the following converse of the Young inequality: Suppose that / and g are continuously differentiable and strictly increasing on [0, oo) and such that /(0) = g(0) = 0, g~l(x) > f(x) for x > 0. If for positive a and b pa pb ab < f(x)dx + / g(x)dx, Jo Jo then / and g are inverse. 1.6.22. Let A=ife K([0,1]) : J f(x)dx = 3, J xf(x)dx = 2 j . Find min / f2(x)dx and a function for which the minimum is attained. 1.6.23. Let A={fe C2([0,1]) : /(0) = /(l) = 0, /'(0) = a} . Find min / {f"(x))2dx and a function for which the minimum is attained. 1.6.24. Does there exist a function / continuously differentiable on [0,2] and such that /(0) = /(2) = 1, \f(x)\ < 1 for x G [0,2] and \fif(x)dx\<n 1.6.25. Suppose / is continuously differentiable on [a, b) and f(a) = f(b) = 0. Show that rb s,ix. :€[a,6] (6 - a)2 7a  1.6. Integral Inequalities 47 1.6.26. Prove the following Holder inequality: If functions /i, /2, • • •, fn are nonnegative and Riemann integrable on [a, 6], and positive real n numbers ai, a2,..., an satisfy ]T a^ = 1, then /•6 n 70 2=1 2 = 1 \/a 1.6.27. Suppose that / and g are nonnegative and Riemann integrable on [a, b]. For p ^ 0 let q be its conjugate, that is, - + ^ = 1. UUllJUgcLLC, L11CLL lO, ~L Using the previous result, prove that (a) if p > 1, then J f(x)g(x)dx< IJ fp(x)dx\ IJ gq(x)dx\ , (b) if p < 0 or 0 < p < 1, and /, g are positive, then / f(x)g(x)dx> (J fp(x)dx) (J g"(x)dx\ . 1.6.28. Suppose that / is continuous on [0,1] and there is a > 0 such that 0 < f(x) < a2/3 for x G [0,1]. Show that if J* f(x)dx = a, then Ji JW)dx > a2/3. 1.6.29. Suppose that / is Riemann integrable on [a, b] and m < f(x) < M. Prove that if (p is continuous and convex on [m, M], then This inequality is called Jensen's inequality. 1.6.30. Prove the following generalization of the Jensen inequality stated in the last problem. Suppose that / and p are Riemann integrable on [a, 6], m < f(x) < M, p(x) > 0 and J p(x)dx > 0. If ip is continuous and convex on [ra, M], then \Jc 1 fb \ 1 r ^ / p(x)f(x)dx I < —^ / p(x)(p(f(x))dx. ^ p(x)dx J a J fa p(x)dx J a  48 Problems. 1: The Riemann-Stieltjes Integral 1.6.31. Let / be Riemann integrable on [0,1] and \f(x)\ < 1, x G [0,1]. Show that J y/\-p{x)dx <Jl-(J f(x)dx) . 1.6.32. Let / be nonnegative and decreasing on [0,1]. Prove that for nonnegative a and 6, > (J x'*if(x)dx\ . 1.6.33. Let / be nonnegative and increasing on [0,1]. Prove that for nonnegative a and 6, 2\ ,i a — b -) J J1x2af(x)dxJ1x2bf(x)dx <( f xa+hf{x)dx^ 1.6.34. Let / be continuous on [a, b] and put f(x) = 0 for x £ [a, b]. For h > 0, we define fh by setting px-\-h Mz) = hL,lt)*- Show that f \fh(x)\dx< f \f(x)\dx. J a J a 1.6.35. Prove the Minkowski inequality for integrals. Assume that A? /2, • • • fn are nonnegative and Riemann integrable on [a, b]. (a) If fc > 1, then U\fi(x) + ...+fn(x))kdx\ < (jUrf(x)dx\ +---+(y"/^)dxj .  1.6. Integral Inequalities 49 (b) If 0 < fc < 1 and /, g are positive, then ' fb \ fe / pb J tf{x)dx\ +•••+(/ fn{x)da pb > 1.6.36. Assume that /i, /2, • • •, /n are nonnegative and Riemann integrable on [a, b]. (a) If k > 1, then / (h(x) + ... + fn(x))kdx> / /^arjdx + • • • + / /£(*)<**. Ja Ja Ja (b) If 0 < fc < 1, then / (fl(x) + ---+fn(x))kdx< f fl<(x)dx + ---+ f ft(x)dx. Ja Ja Ja 1.6.37. Prove the following inequality of Steffensen: If p is Riemann integrable on [a, b] and 0 < g{x) < 1 for every x G [a, 6], and / decreases on that interval, then /*fr pb ra-j-X / /0)dx < / f(x)g(x)dx < I f(x)dx, Jb—X Ja Ja where fb A = / q(x)dx. Ja 1.6.38. Prove the following Bellman generalization of Steffensen's inequality: If g is Riemann integrable on [0,1] and 0 < g(x) < 1 for every x G [0,1], and / is nonnegative and decreasing on that interval, then for p > 1 J f(x)g(x)dx) < J (f(x))*dx, r I g(x)dz Jo where p  50 Problems. 1: The Riemann-Stieltjes Integral 1.6,39. Using 1.6.38, show that if g is Riemann integrable on [a, b], 0 < g(x) < 1 for every x E [a, 6], and if / is nonnegative and decreasing on that interval, then for p > 1 p (b where -^(jj(X)g(X)dx] <£+\f{x))*dx, 1 / fb {b-a)p~i yja yK) v lx I . 1.6.40. Prove the following variant of Steffensen's inequality due to R. Apery. Let / be a nonnegative and decreasing function on [0, oo) and let g be a function defined on [0, oo) such that 0 < g(x) < A (A > 0) with J0°° g(x)dx < oo. Then /»oo /»A / f(x)g{x)dx <A f(x)dx, Jo Jo where 1 f°° X = ~A / 9^dx' 1.6.41* Prove the following Bellman generalization of Steffensen's inequality. Let / : [a, b] —» R be a nonnegative decreasing function and let g : [a, b] —> R be a nonnegative and Riemann integrable function such that 0<g(x) If g(t)dt\ <l, xe [a, b]. If p > 1, then ra+A where U g(x)f(x)dx\ <jf (f(x))*dx, A = I / #(»Gfo I . Moreover, prove that if 0 < p < 1, then ( fg(x)f(x)dx) > f {f{x)fdx \Ja I Jb-X  1.6. Integral Inequalities 51 with A defined above. 1.6.42. Let g\ and g2 be functions integrable on [a, b] such that for every x G [a, 6] px px / gi(t)dt> / g2{t)dt J a J a and pb pb / 0i(«)<**= / g2(t)dt, J a J a Show that if / is increasing on [a, b], then /•fr pb / f(t)9l(t)dt < / f(t)g2(t)dt, J a J a and if / is decreasing on [a, 6], then / f(t)gi(t)dt> f f(t)g2(t)dt. J a J a 1.6.43. Use the Steffensen inequality to prove that if / is continuously differentiable on [a, b] and m < f'(x) < M (m < M) for x G [a, 6], then (M - m)A2 /(b) - /(a) (M - m)(ft - a - A)2 + (ft-a)2 - 6-a " (fe-a)2 where f(b)-f{a)-m(b-a) A M -m 1.6.44. Prove that if / is continuously differentiable on [a, b] and m < f'(x) < M (m < M) for x G [a, fe], then / f(x)dx- J a f(a)+f(b) (b-a) < 2 (/(6) - /(a) - m(b - a))(M(b - a) - f(b) + /(a)) 2(M-ra) 1.6.45. Prove the following Opza/ inequality: If / is continuously differentiable on [0, a] and /(0) = 0, then £\f(x)f'(x)\dx<lj\f'(x))2dx.  52 Problems. 1: The Riemann-Stieltjes Integral 1.6.46. Prove the following generalization of Opial's inequality. If / e Cn([0,a]) is such that /(0) = /'(0) = • • • = /("-^(O) = 0, n > 1, then f \f{x)f(n-l){x)\dx < y J\fM(x))2dx. 1.6.47. Prove this generalization of Opial's inequality. If / is continuously differentiate on [0, a], /(0) = 0 and p > 0, q > 1, then fa\f(x)\P\f\x^dx<^- fa\f'{x)\*+*dx. Jo p + q Jo 1.6.48. Suppose / G C2n([a,fr]) is such that f^(a) = /<*>(&) = 0 for fc = 0,1,... ,n - 1. Prove that if M = max |/^2n)(x)|, then x£[a,b] / /(a:)da: 1.7. Jordan Measure For a^ < bi, i = 1, 2,... ,p, the volume of the interval R = [ai, &i] x [a2,&2] x- • -x [ap, bp] is the number |R| = (b\ — ai)(p2 — 02). •. (bp —ap). In the case p = 1 or p = 2, the number |R| is called the length and the area of R, respectively. We say that two intervals are separate if they have at most boundary points in common. A union of finitely many intervals is called an elementary set. Any elementary set E can be expressed (in infinitely many ways) as a union of pairwise separate intervals Ri,..., Rn. Then the volume of the elementary set E is the number |E| = |Ri|H h|Rn|- The volume |E| does not depend on the choice of the R^. For a nonempty set A C Mp, we put | A|* = sup |E|, where the supremum is taken over all elementary sets E contained in A. The number |A|* is called the inner volume (or the Jordan inner measure) of A. If A is unbounded, its inner Jordan measure may be infinite. If A is bounded, then its outer volume (or the Jordan outer measure) is the number |A|* = inf |E|, where the infimum is taken over all elementary sets E containing A. If A is unbounded, then |A|* = lim |AnRfc|*, where Kk = \-k,k] x ••• x [-fc,fc] C Rp. If |A|* = |A|* = |A|, we say that A has volume |A| (or is Jordan measurable and its Jordan measure is |A|). In the case when |A| = < (n!)2(fr- (2n)!(2n + l)! -M.  1.7. Jordan Measure 53 +00 we assume additionally that A fl R is Jordan measurable for any interval R. Moreover, we assume that |0| =0. 1.7.1. Show that a bounded set A is Jordan measurable if and only if, given e > 0, there are two elementary sets Ei and E2 such that Ei c A C E2 and |E2| - |Ei| < e. 1.7.2. Show that if Ai,..., An are bounded, Jordan measurable and pairwise separate, that is, A° fl A° = 0 for i 7^ j, then |AiU---UAn| = |Ai| + .-- + |An|. 1.7.3. Show that if Ai and A2 are Jordan measurable, then so are Ai U A2 and Ai fl A2, and |AiUA2| + |AinA2| = |Ai| + |A2|. 1.7.4. Show that if Ai and A2 are Jordan measurable, Ai C A2, and I Ai| < +00, then A2 \ Ai is Jordan measurable and |A2 \ Ai| = |A2|-|Ai|. 1.7.5. Show that the set A = {(x,y)GR2:xe[0,l], ye[0,l]\Q} is not Jordan measurable. 1.7.6. Show that the set B - {(*, y)eR2:xe [0,1], y G [0,1] \ {1/n : n G N}} is Jordan measurable. 1.7.7. Show that if |B|* = 0, then for any bounded A, |AUB|* = |A|*. Give an example to show that |B|* = 0 is not sufficient for this equality to hold. 1.7.8. Show that if A is Jordan measurable, then its interior and closure are Jordan measurable and |A| = |A°| = |A|. 1.7.9. Show that A is Jordan measurable if and only if the Jordan measure of its boundary dA is zero.  54 Problems. 1: The Riemann-Stieltjes Integral 1.7.10. Prove that the assumption that Ai,..., An are bounded can be dropped from the assertion given in 1.7.2. 1.7.11. Let C C [0,1] denote the Cantor set (for the definition of the Cantor set see, e.g., the solution to II, 1.3.1). Show that |C| = 0. 1.7.12. Let A be a Cantor-like set defined as follows. Given a G (0,1), we remove from [0,1] an open interval (| — f > | + f) and denote by Ei the union of the two remaining closed intervals. Next we remove the open middle intervals of length a/23 of the two intervals constituting Ei and denote by E2 the union of the four remaining closed intervals. We repeat the process with each of the four intervals, removing the open middle intervals of length a/25. Continuing the process, we obtain the sequence {En} of sets, where En is the union of 2n closed intervals, and we put A = H^Li En- Show that A is not Jordan measurable. 1.7.13. Is A = {1 + ^-, 2 + ^-, • • • : n G N} a Jordan measurable subset of R? 1.7.14. Let {rk} be the sequence of all rationais contained in [0,1] and let I& be an open interval centered at r^, k G N, of length l/2fc+1. Show that the set A = (JfcLi *fc *s not Jordan measurable. 1.7.15. Give examples of nonmeasurable open sets and nonmeasur- able closed sets. 1.7.16. Let A C [a, b]. If xa is the characteristic function of A, then pb pb |A|* = / \A(x)dx and |A|* = / xa(x)(Ix. J a J a 1.7.17. For a nonnegative and bounded function / on an interval [a, b], put A/ = {(x,y) eR2 : a < x < b,0 <y < f(x)}. Prove that rb rb lA/l* = / f(x)dx, and |A/|* = / f(x)dx. J a J a 1.7.18. Let / be a bounded function on [a,b]. For e > 0 define J£ = {x G [a, b],Of(x) > e}, where o/(x) denotes the oscillation of / at x (see, e.g., II, 1.7.12). Prove that / is Riemann integrabie on [a, b] if and only if | Je| =0 for every e > 0.  1.7. Jordan Measure 55 1.7.19. Show that a bounded set A is Jordan measurable if and only if there exist two sequences {Bn} and {Cn} of Jordan measurable sets such that Bn C A C Cn and lim |Bn| = lim |Cn| = |A|. n—>oo n—>oo 1.7.20. Show that the set A = I (x, y) G R2 : 0 < x < 1, 0 < y < is Jordan measurable. 1.7.21. Set A- {(x,y) G M2 : (x - l)2 + y2 < l} and Show that the set A \ (J^Li Kn is Jordan measurable. 1.7.22. Derive the following formula for the Jordan measure of sets defined by inequalities in the polar coordinates (r, 6). If / is a non- negative and continuous function on [a,/?], f3 — a < 2tt, then the set A = {(r, 6) :a<6<[3,0<r< f(0)} is Jordan measurable, and its area is \A\ = \Jj\e)de. 1.7.23. Find the area of one of the congruent loops of the lemniscate r2 = a2 cos(2#), where a > 0 is fixed. 1.7.24. The curve r = a(l + cos(30)), a > 0, consists of three congruent leaves, tangent to each other at the origin. Find the area of one of the leaves. 1.7.25. Find the area of the limagon r = a + bcosO, where positive numbers a and b are given, distinguishing the cases in which a > 6, a — b, and a < b. 1.7.26. Find the area of the loop of the curve x5 + y5 = 5ax2y2, where a > 0 is given. 1.7.27. Find the area of the region that lies within the limagon r = 1 + 2cos# and outside the circle r = 2. . 1 sin — x  56 Problems. 1: The Riemann-Stieltjes Integral 1.7.28. Suppose / is nonnegative and continuous on [a, b}. A plane region A/ = {(x,y) G M2 : a < 6, 0 < y < f(x)} is revolved about the x-axis, generating a solid of revolution V with volume |V|. Prove that |V|=7r/ f(x)dx. J a 1.7.29. Suppose / is nonnegative and continuous on [a, b] and 0 < a. Prove that the volume of the solid V generated by the plane region A/ = {(x, y) G IR2 : a < b, 0 < y < f(x)} when it revolves about the y-axis is given by |V| = 2?r / xf(x)dx. J a 1.7.30. Find the volume of a torus obtained by revolving the disk {(#, y) G IR2 : (x — a)2 +y2 < r2}, where 0 < a < r, about the y-axis. 1.7.31. Find the volume of a solid obtained by revolving an unbounded region under the graph of f(x) = e_xv/sinx over the set UfcLoI^71"' (2& + l)7r] about the x-axis. 1.7.32. Show that the length L of the ellipse satisfies 7r(a + 6) < L < 7rv/2(a2 + 62). 1.7.33. Show that the length of the ellipse (f)\(f)' = l, «>-«, is given by V ^V (2n)!! ^ 2»-V where e = ~b is the eccentricity of the ellipse. 1.7.34. Derive the following formula for the length of a curve in the polar coordinates (r, 6). If / is continuously differentiable on [a,/?], then the length L of the curve r = f{0), a < 6 < /?, is given by  1.7. Jordan Measure 57 1.7.35. Find the length of the curve with polar equation Q (a) r = asin3-, 0 < 0 < 3tt. o (b) r=~, l—^ ~1<0<1- v J 1 + cos 0 2 " " 2 1.7.36. Find the length of the curve with polar equation 0=\(r+l), l<r<3. 1.7.37. Find the length of the curve r=l + cos£, 0 = *-tan-, 0 < t < /?, where (r, 0) are polar coordinates and 0 < (3 < n.  This page intentionally left blank  Chapter 2 The Lebesgue Integral 2.1. Lebesgue Measure on the Real Line For A C R, we consider a countable collections of closed intervals oo {In}, In = [an, bn],an < bni that cover A, that is, A C (J In and we 71 = 1 define the Lebesgue outer measure m* (A) of A by oo oo ra*(A) = inf ^ |In| = inf ^(6n - an), 71=1 71=1 where the infimum is taken over all countable coverings of A by closed intervals. Clearly, the closed intervals in the definition of the Lebesgue outer measure can be replaced by open intervals (an,6n). The Lebesgue outer measure enjoys the following properties: (i) If A C B, then ra*(A) < ra*(B). / oo \ oo (ii) m* MJ An < £m*(An). \n=l / n=l A set A is said to be Lebesgue measurable (or measurable), if for each set E C R the Caratheodory condition ra*(E) = m*(E n A) + m*(E \ A) is satisfied. The set Wl of all Lebesgue measurable subsets of R is a o- algebra; that is, the complement of a measurable set is measurable and the union of a countable collection of measurable sets is measurable. 59  60 Problems. 2: The Lebesgue Integral Recall that the collection *B of Borel sets is the smallest cr-algebra which contains all of the open sets in R. Every Borel set is measurable. If A is a measurable set, we define the Lebesgue measure m(A) to be the outer measure of A. The measure m is countably additive on OJt; that is, if the sets An are pairwise disjoint, then (oo \ oo (J An I = ^m(An). 71=1 / 71=1 2.1.1. Show that if m*(A) = 0, then A is measurable. 2.1.2. Let S = (Jri=i In> where In, n G N, are closed intervals. Show that m(S) = |S|*, where |S|* denotes the inner Jordan measure. 2.1.3. Let Sx = Ur=i In and S2 = U~=i Kn, where In, Kn, n G N, are closed intervals. Show that m(S1 U S2) + m(Si n S2) = ra^) + m(S2). 2.1.4. Show that for any two sets A and B, ra*(A U B) + m*(A fl B) < m*(A) + m*(B). 2.1.5. For an open set G, let A C G and B n G = 0. Show that m*(A U B) = m*(A) + m*(B). 2.1.6. Suppose that A and B have positive distance, that is, d(A, B) = inf {|x -y|:xGA,yeB}>0. Then m*(A U B) = m*(A) + m*(B). 2.1.7. Show that if ra(A) = 0, then for any B, m*(AuB) = m*(B\A) = m*(B). 2.1.8. Show that if A is a measurable set of finite measure, then for any B D A, m*(B\A)=m*(B)-m(A). 2.1.9. Show that if A and B are measurable, then ra(A U B) + ra(A n B) = ra(A) + m(B). 2.1.10. Show that if m*(A) < oo and there is a measurable subset Ai of A such that ?n(Ai) = m*(A), then A is measurable.  2.1. Lebesgue Measure on the Real Line 61 2.1.11. Let A be any set in R. Show that for each e > 0 there is an open set G such that A C G and m(G) < m*(A) +e. Show also that there is a Qs set A2 such that A C A2 and m(A2) = m*(A). 2.1.12. Prove that for A C R the following statements are equivalent: (i) A is measurable. (ii) Given e > 0, there is an open set G D A such that m*(G\A) <e. (hi) There is a Q6 set U D A such that ra*(U \ A) = 0. (iv) Given s > 0, there is a closed set F C A such that ra*(A\F) <e. (v) There is an Ta set V C A such that ra*(A \ V) = 0. 2.1.13. Show that if m*(A) < oo, then A is measurable if and only if for each e > 0 there is a finite union W of open intervals such that ra*(W A A) < £, where W A A is the symmetric difference for W and A, that is, W A A = (W \ A) U (A \ W). 2.1.14. For A C R, define the Lebesgue inner measure m*(A) of A by setting ra*(A) = sup{ra(B) : B G S0t, BcA}, where 971 denotes the a-algebra of all measurable subsets of R. Prove: (a) If A is measurable, then rn*(A) = ra*(A). (b) If A is a subset of a bounded closed interval I, then m*(A) = |I|-m*(I\A). (c) If m*(A) = m*(A) < oo, then A is measurable. (d) For any sets A and C, m*(AuC) + ra*(AnC) > m*(A) +7nsK(C). (e) If An, n = 1,2,..., are pairwise disjoint, then m* ( |J An J > ^m^A^). \n=l / n=l (f) If M is of measure zero, then m*(A U M) = ?n*(A).  62 Problems. 2: The Lebesgue Integral 2.1.15. Prove that A C I, where I is a bounded and closed interval, is measurable if and only if |I|=m*(A)+m*(I\A). 2.1.16. Let A and B be given sets of finite outer measure. Show that ra* (A U B) = ra* (A) + ra* (B) if and only if there are measurable sets Ai and Bi such A C Ai, B C Bi and ra(Ai n Bi) = 0. 2.1.17. Let A and B be given sets of finite outer measure. Show that if ra*(AuB) = ra*(A)+ra*(B), thenra*(AuB) = ra*(A) + ra*(B). 2.1.18. Prove that if {An} is an increasing sequence of measurable sets, then ra \n=l I J An | = lim ra(An). ^^ / n—»oo 2.1.19. Prove that if {An} is a decreasing sequence of measurable sets and m(Afc) is finite for at least one k, then Pi An I = lim m(An). 1 ' / 71—KX) ra \n=l / Show also that the assumption that m(Afc) < oo for some k cannot be omitted. 2.1.20. For a sequence {An} of sets in R, we define the limit superior and the limit inferior of {An} by setting oo oo oo oo lim An = P| (J An and lim An = (J Q An. Tl ^OO j-^—»O0 k=\n=k k=\n=k (a) Show that if An, n G N, are measurable, then m ( lim An ) < lim m(An). \n^oo / n^oo (b) Show that if, moreover, m(An U An+i U ...) < oo for at least one n, then m( lim An) > lim m(An). Vn^oo / n—>oo 2.1.^1. We say that a sequence {An} of sets in R converges if lim An n—>-oo = lim An, and we denote the common value by lim An.  2.1. Lebesgue Measure on the Real Line 63 (a) Show that a monotonic sequence of sets converges. (b) Show that, if a sequence {An} of measurable sets, An C B, where m*(B) < oo, converges, then m ( lim An ) = lim ra(An). \n—»oo / n—»oo 2.1.22. Show that the Lebesgue measure of the set A defined in 1.7.12 is equal to 1 — a. 2.1.23. Let A be the set of points in [0,1] such that x is in A if and only if the decimal expansion of x does not require the use of the digit 7. Show that A has Lebesgue measure zero. 2.1.24. Let B C R be the set of all numbers whose decimal expansions do not require the use of the digit 7 after the decimal point. Show that B has Lebesgue measure zero. 2.1.25. Let A be the set of points in [0,1] which admit of binary expansions with zeroes in all even positions. Show that A is a nowhere dense set of Lebesgue measure zero. 2.1.26. Find the Lebesgue measure of the set of points in [0,1] which admit decimal expansions containing all the digits 1, 2,..., 9. 2.1.27. What is the Lebesgue measure of the set of points in [0,1] which admit decimal expansions O.diG^cfe ... such that no sequence <^3fc+i^3/c+2^3/c+3 consists of three consecutive 2's? 2.1.28. Let A be the union of intervals centered at points of the Cantor set and each of length 0.1. Find the Lebesgue measure of A. 2.1.29. Show that if A is a bounded measurable set of measure m(A) = p > 0, then for each q G (0,p) there is a measurable set B C A of measure q. 2.1.30. Show that if 0 < 772(A) < 00, then for each positive q < 771(A) there is a measurable set B C A of measure q. 2.1.31. Show that if 0 < m(A) < 00, then for each positive q < m(A) there is a perfect set B C A of measure q. 2.1.32. Show that any set A of positive Lebesgue measure has the cardinality of the continuum.  64 Problems. 2: The Lebesgue Integral 2.1.33. Show that any nonempty and closed set A C R of Lebesgue measure zero is nowhere dense. 2.1.34. Suppose that A C R is a nowhere dense set of Lebesgue measure zero. Must its closure be of Lebesgue measure zero? 2.1.35. Show that if A C [a, b] and m(A) > 0, then there are x and y in A such that \x — y\ is an irrational number. 2.1.36. Does there exist a countable collection of nowhere dense and perfect subsets of [0,1] whose union is of Lebesgue measure 1? 2.1.37. Does there exist a nowhere dense and perfect subset of [0,1] of Lebesgue measure 1? 2.1.38. Give an example of a measurable set A C R with the following property: For each interval (a, /?), ra(An(a,/?)) >0 and ra((R \ A) n (a,/?)) > 0. 2.1.39. Assume that a measurable set A C R has the property that for each 8 > 0, m(Afl (—8, 8)) > 0 and 0 G A. Prove that there exists a perfect set B C A such that ra(B n (—8, 8)) > 0 for every 8 > 0. 2.1.40. A measurable set A C R is said to have density d at x if the limit ra(A n \x — h,x + hX) hm — — h^o+ 2h exists and is equal to d. If d = 1, then x is called a point of density of A, and if d = 0, then x is called a point of dispersion of A. Find the points of density and points of dispersion of A = (—1, 0)u(0,1)U{2}. 2.1.41. Given a G (0,1), construct a set A whose density at xq G R is equal to a. 2.1.42. Let A be a measurable set such that 0 G A is a point of density of A. Prove that there is a perfect set B C A such that 0 is a point of density of B. 2.1.43. If x and y are in [0,1), we define the sum x + y(mod 1) to be x + y, if x + y < 1, and to be x + y - 1, if x + y > 1. For A C [0,1) and a G [0,1), we define the translate modulo 1 of A to be the set A + a(mod 1) = {x + a (mod 1) : x G A}. Show that m*(A) = m*(A + a(mod 1)).  2.1. Lebesgue Measure on the Real Line 65 2.1.44. Show that if A C [0,1) and a G [0,1), then the set A + a(mod 1) is measurable if and only if A is measurable and ra(A + a(mod 1)) = m(A). 2.1.45. We say that x, y G [0,1) are equivalent, and write x ~ y, if and only if x — y is a rational. This equivalence relation partitions [0,1) into an uncountable family of disjoint equivalence classes. By the axiom of choice there is a set V which contains exactly one element from each equivalence class. We call any such a set a Vitali set Prove that a Vitali set is nonmeasurable. 2.1.46. Show that if A is a measurable subset of a Vitali set V, then ra(A) = 0. 2.1.47. Show, by example, that the converse of the result stated in 2.1.17 is not true; that is, there exist sets A and B such that ra*(AUB) = ra*(A)+ra*(B) but ra*(AuB) < ra*(A) + ra*(B). 2.1.48. Show that any set of positive outer measure contains a non- measurable subset. 2.1.49. Give an example of a sequence {An} of pairwise disjoint sets such that (oo \ oo (J An <^m*(An). 71=1 / 71=1 2.1.50. Give an example of a decreasing sequence {An} of sets such that m* (Ai) < oo and ra* I P| An I < lim ra*(An). \n=l / 2.1.51. Show that if A is a measurable set of positive measure, then there is 8 > 0 such that An (A + x) is nonempty whenever \x\ < S. 2.1.52. Let Vfc, k = 0,1,..., be the sets defined in the solution to 2.1.45. Show that each of the sets An = (Jfc=o Vfc is nonmeasurable. 2.1.53. Suppose that A C R is a measurable set for which there is a positive c such that m(A Hi) > c|I| for every interval I. Show that the complement of A is of measure zero.  66 Problems. 2: The Lebesgue Integral 2.1.54. Prove that there is a set A C R such that each Lebesgue measurable set that is included in A or in Ac has Lebesgue measure zero. 2.1.55. Prove the following Lebesgue criterion for Riemann integra- bility: A bounded function on a closed bounded interval is Riemann integrable if and only if the set of discontinuities of / has Lebesgue measure zero. 2.1.56. For / : [a, b] —► R, let D denote the set of discontinuities of / and L the set of points where / has a left-hand limit. Show that D H L is countable. Conclude the following criterion for Riemann integr ability: a bounded function on [a, b] is Riemann integrable if and only if the set [a, b] \ L has Lebesgue measure zero. 2.1.57. Let / : [0,1] -+ R be continuous with /(0) = /(l) = 0. Show that the Lebesgue measure of A = {he [0,1] : f(x + h) = f(x) for some x G [0,1]} is at least 1/2. 2.2. Lebesgue Measurable Functions An extended real-valued function / : A —» R defined on a measurable set A C R is said to be Lebesgue measurable (or measurable for short) on A if /_1((c, oo]) = {x G A : f(x) > c} is a Lebesgue measurable subset of A for every real number c. We have Theorem 1. The following statements are equivalent: (i) / is Lebesgue measurable on A. (ii) /_1([c, oo]) = {x G A : f(x) > c] is a Lebesgue measurable subset of A for every real c. (hi) /_1([—oo,c)) = {x G A : f(x) < c} is a Lebesgue measurable subset of A for every real c. (iv) /_1([—oo,c]) = {x G A : f(x) < c] is a Lebesgue measurable subset of A for every real c. We will also use the following.  2.2. Lebesgue Measurable Functions 67 Theorem 2. Let f and g be measurable real-valued functions defined on A, let F be a real-valued function continuous on R2? and put h{x) = F(f(x),g(x)) for x G A. Then h is measurable on A. In particular, f + g and fg are measurable. Theorem 3. Let {fn} be a sequence of measurable functions on A; then hi(x) = sup{fn(x) : n G N} and h2(x) = limn^00fn(x) are also measurable on A. In particular, iff and g are measurable on A, then so are max{/,^} and min{/, g}. A property is said to hold almost everywhere (abbreviated a.e.) on a measurable set if the set of points where it fails to hold is a set of measure zero. In particular, we say that f — g a.e. if / and g have the same domain A and m({x G A : f(x) ^ g(x)}) = 0. Suppose A = UlLi^> where the sets A; are measurable and pair wise disjoint. A function tp defined on A by n i=\ where Ci, C2,..., cn are real numbers, is said to be a simple function. The following theorem is called the approximation theorem for measurable functions. Theorem 4. For every measurable function f defined on a measurable set A, there is a sequence {tpn} of simple functions which converges pointwise to f. In case f is nonnegative, the sequence {^n} may be constructed so that it is monotonically increasing. In case f is bounded on A, the sequence {^n} may be chosen so that the convergence is uniform on A. 2.2.1. Show that if / is measurable, then so is |/|. Does the measur- ability of |/| imply the measurability of /? 2.2.2. Give examples of nonmeasurable functions / and g such that (a) / + 9 1S measurable, (b) fg is measurable. 2.2.3. Let / be a real-valued function defined on R. Show that the measurability of the set {x : f(x) = c] for every real c is not sufficient for / to be measurable.  68 Problems. 2: The Lebesgue Integral 2.2.4. Let C be a dense subset of R. Show that a function / defined on a measurable set A is measurable if and only if {x G A : f(x) > c] is measurable for every c G C. 2.2.5. Show that a real-valued function / defined on a measurable set A is measurable if and only if /_1(G) is measurable for every open GcR. 2.2.6. Show that if a real-valued function / defined on R is measurable, then /_1(B) is measurable for every Borel set B C R. 2.2.7. Prove that continuous real-valued functions defined on measurable sets are measurable. 2.2.8. Suppose extended-valued functions / and g are defined on a measurable set A. Prove that if / is a measurable function and f = g a.e., then g is measurable. 2.2.9. Prove that every Riemann integrable function defined on [a, b] is measurable on [a, b]. 2.2.10. Show that each function of bounded variation on [a, b] is measurable. 2.2.11. Define the Cantor function tp : [0,1] —► [0,1] as follows: If oo x is an element of the Cantor set C and x = Y2 f£ with an = 0 or 71=1 an = 2, then we put (oo \ oo 1 / v 371 / / j 2 2n 71=1 / 71=1 that is, if an is the nth ternary digit for x, then the nth binary digit for (f(x) is an/2. We extend tp to [0,1] by setting ip{x) = sup{(/?(y) : y G C, y < x). Show that the Cantor function maps the Cantor set C onto [0,1]. Show also that ip is increasing and continuous on [0,1] and tp''(x) = 0 a.e. on [0,1]. 2.2.12. Prove that if / is a one-one continuous mapping of R onto R, then / maps Borel sets onto Borel sets. 2.2.13. Give an example of  2.2. Lebesgue Measurable Functions 69 (a) a measurable function / and a measurable set A such that /(A) is nonmeasurable, (b) a measurable function g and a measurable set B such that #-1(B) is nonmeasurable. 2.2.14. Give an example of a measurable set that is not Borel. 2.2.15. Assume that / is continuous on [a, b}. Show that / satisfies the condition E C [a, b] and ra(E) = 0 implies ra(/(E)) = 0 if and only if for any measurable Ac [a, b] its image /(A) is measurable. 2.2.16. Suppose that a real-valued function g is measurable on A and / is continuous on g(A). Show that / o g is measurable. 2.2.17. Suppose that g is continuous on [a, b] and h is real-valued and measurable on p([a, b]). Must ho g he measurable? 2.2.18. Suppose that a real-valued function g is measurable on A and / defined on R is such that for every open set G, the inverse image /_1(G) is a Borel set. Show that / o g is measurable. 2.2.19. Give an example of a measurable function whose inverse is not measurable. 2.2.20. Let / be a function differentiate on [a, b]. Show that f is measurable on [a, b}. 2.2.21. Let A C R be a measurable set of finite measure and / a measurable function on A which is finite almost everywhere. Then, given s > 0, there is a measurable set B C A such that m( A \ B) < e and / restricted to B is bounded. 2.2.22. Prove the following Egorov theorem. Let A C R be a measurable set of finite measure. If {fn} is a sequence of measurable functions which converges to a real-valued function / almost everywhere on A, then, given e > 0, there exists a measurable subset B of A such that m(A\B) < e and the sequence {fn} converges uniformly to / on B.  70 Problems. 2: The Lebesgue Integral 2.2.23. Show by example that the assumption m(A) < oo is essential in the Egorov theorem. 2.2.24. Suppose that A is measurable and {fn} is a sequence of measurable functions which converges to / almost everywhere on A. Prove that there is a set B C A such that B = (J^i Bi> m(A\B) = 0 and the sequence {fn} converges uniformly to / on every B^. 2.2.25. Let {Vn} be the sequence of sets defined in the solution to 2.1.45 and let {fn} be the sequence of functions on [0,1) defined by fn = X|j^ Vi- Show that fn —» 0 on [0,1), but the assertion of the Egorov theorem does not hold, that is, there is e > 0 such that on any measurable subset B of [0,1) with m([0,1) \ B) < e the convergence is not uniform. 2.2.26. Construct a sequence {fn} of measurable functions on [0,1] such that the sequence converges everywhere on [0,1] but for every set B C [0,1] of measure 1, the sequence fails to converge uniformly on B. 2.2.27. Prove the following Lusin theorem: In order that a real- valued function / defined on a measurable set A be measurable, a necessary and sufficient condition is that for every e > 0 there is a closed set F C A such that m(A \ F) < e and / restricted to F is continuous. 2.2.28. Using the result in 2.1.38, construct a function /, measurable on R, such that for any set E of measure zero, / is not continuous at any point in R \ E. 2.2.29. Let / : [0, a] —» R be a measurable function. Show that there exists a monotone decreasing function g on [0, a] such that for any real y, m{{x G [0,a] : f{x) > y}) = m({x G [0, a] : g{x) > y}). 2.2.30. Let {fn} be a sequence of real-valued functions measurable on A. We say that {fn} converges in measure to a measurable function f if for every e > 0, lim m({x G A : \fn(x) — f(x)\ > s}) = 0. n—^-oo Show that if a sequence {fn} converges in measure to / and {fn} converges in measure to #, then / = g a.e. on A.  2.3. Lebesgue Integration 71 2.2.31. Prove the following theorem of Lebesgue. If m(A) < oo and {fn} converges to / a.e. on A, then {fn} converges in measure to /. 2.2.32. Show by example that the assumption m(A) < oo is essential in the Lebesgue theorem stated above. 2.2.33. Give an example of a sequence of measurable functions on the interval [0,1] that converges in measure on [0,1] but is not convergent at any point of that interval. 2.2.34. Let {fn} be the marching sequence defined in the solution to the previous problem. Find a subsequence of it that converges to the zero function a.e. on [0, lj. 2.2.35. Prove the following Riesz theorem. Every sequence {fn} convergent in measure to / on A contains a subsequence converging to / a.e. on A. 2.2.36. Let {fn} be a sequence of monotonically increasing functions on (a, b). Show that if the sequence converges in measure to /, then lim fn(x) = f(x) at each point x of continuity of /. n—>oo 2.2.37. Prove the following Frechet theorem. If / is a real-valued measurable function on A, then there is an Ta set H such that ra(A \ H) = 0 and / restricted to H is in the first Baire class (that is, / is a pointwise limit of a sequence of continuous functions on H). 2.2.38. Prove the following Vitali theorem. If / is a real-valued measurable function on A, then there is a function g in the second Baire class (that is, g is a pointwise limit of a sequence of functions in the first Baire class on A) such / = g a.e. 2.3. Lebesgue Integration n The Lebesgue integral of a simple function ip(x) = Yl ci\A7{x) on 2=1 A, where A = UlLi ^^ an<^ tne ^ arG P^irwise disjoint measurable sets, is defined as r n 7A r=l  72 Problems. 2: The Lebesgue Integral If / is measurable and nonnegative on A, we define / fdm = sup / (pdm, J a J a where the supremum is taken over all simple functions <p such that 0 < ip < /. The function / is said to be integrable (or summable) on A in the Lebesgue sense if its integral over A is finite. If / is measurable on A, then we define / fdm = / f+dm — I f~ dm, J A J A J A where /+ = max{/, 0} and /~ = — min{/, 0}. We say that / is integrable on A if /+ and /~ are integrable on A. Let p be a positive real number. A measurable function / defined on A is said to belong to the space LP(A) if JA \f\pdm < oo. In this case we will write ||/||p = (/ \f\Pdm)1/p. In the case when A = [a, b] we write / G Lp[a,b\. A measurable function / defined on A is said to be essentially bounded if m({x G A : \f(x)\ > r}) = 0 for some real number r. In this case we define the essential supremum of / by ll/IU = inf{r : m({x G A : \f(x)\ > r}) = 0}. Then the set E = {x G A : \f(x)\ > ||/||oo} is of measure zero and \f(x)\ < ll/IU outside E. Thus \f(x)\ < Wf]^ a.e. We will often use the following theorems: Theorem 1 (Lebesgue's Monotone Convergence Theorem). Suppose {fn} is an increasing sequence of nonnegative measurable functions on A. If f(x) = lim fn(x), xGA, then n—>oo Am / fndm = / fdm. A JA Theorem 2 (Fatou's Theorem). Suppose {fn} is a sequence of nonnegative and measurable functions on A. Then / 1™ fndm < lim / fndm. J A n—>oo n—>oo J A  2.3. Lebesgue Integration 73 Theorem 3 (Lebesgue's Dominated Convergence Theorem). Suppose {fn} is a sequence of measurable functions on A and f(x) — lim fn(x), x £ A. If there exists a function g integrable on A and n—>oo such that \fn(x)\ < g(x), n = 1, 2,..., x G A, then lim / fndm = / fdm. J A J A n—>oo Theorem 4. If f is Riemann integrable on [a,b], then f is Lebesgue integrable and / f{x)dx = / fdm. Ja J[a,b] 2.3.1. Find the Lebesgue integral of the function / defined by setting f( . jx2 if xG[0,l]\Q, \i if xg [0,i]nQ. Is the function Riemann integrable on [0,1]? 2.3.2. Let / be defined on [0,1] as follows: f(x) = 0 if x is an element of the Cantor set C and f(x) = n on each removed interval of length l/3n (see, e.g., the solution to II, 1.3.1 for the definition of C). Find /[o,i] fdm- 2.3.3. Let C denote the Cantor set. Find f,Q ^ fdm, where /(*) sin(7rx) if x e [0,1/2] \ C, cos(ttx) if x e [1/2,1] \ C, x2 if xec. 2.3.4. Prove that if / is Lebesgue integrable on A, then, given e > 0, there is S > 0 such that JB \f\dm < e whenever B C A and m(B) < 5; that is, the Lebesgue integral is absolutely continuous with respect to the Lebesgue measure. 2.3.5. Show that if / is a Lebesgue integrable function on A and An = {x G A : \f(x)\ > n}, then lim n • m(An) = 0. n—>oo 2.3.6. Show that if / > 0 on a set A of positive measure and JA fdm = 0, then / = 0 a.e. on A.  74 Problems. 2: The Lebesgue Integral 2.3.7. Show that if JB fdm = 0 for every measurable subset B of A, m(A) > 0, then / = 0 a.e. on A. 2.3.8. Show that if / is Lebesgue integrable on A and fdm\ = / \f\dm, A I J A then either / > 0 a.e. on A or f < 0 a.e. on A 2.3.9. Prove that if {/n} is a sequence of nonnegative and measurable functions on A such that lim fA fndm = 0, then {fn} converges to n—>oo A 0 in measure. Show by example that convergence in measure cannot be replaced by a.e. convergence. 2.3.10. For a sequence {fn} of measurable functions on a set A of finite measure, show that lim / T^rndm = 0 n~>0° J A 1 + |/n| if and only if {fn} converges to 0 in measure. Show by example that the assumption m(A) < oo cannot be omitted. 2.3.11. Suppose / is nonnegative and measurable on a set A of finite measure. Prove that / is Lebesgue integrable on A if and only if oo the series ]T km(Ak), where A& = {x G A : k < f(x) < k + 1}, k=0 converges. 2.3.12. Suppose / is nonnegative and measurable on a set A of finite measure. Prove that / is Lebesgue integrable on A if and only if the oo series ^ 77^(B/C), where B& = {x G A : f(x) > /c}, converges. 2.3.13. Suppose / is nonnegative and integrable on a set A of finite measure. For e > 0, define {x e A:ne < f(x) < (n + l)e}. fdm. A oo S(e) = y^jiem(An), n=0 Prove that where A lim S(e)  2.3. Lebesgue Integration 75 2.3.14. Let {fn} be a sequence of nonnegative functions converging to / on R, and suppose that lim fR fndm = JR fdm < oo. Show that n—>oo for each measurable set A lim / n->ooJA fndm = / fdm. A JA 2.3.15. Suppose that {fn} is a sequence of real-valued functions in- tegrable on a set A of finite measure. Show that if the sequence is uniformly convergent on A, then lim / fndm = / lim fndm. n~>oc J A J A n~>°° /A JA ' 2.3.16. Let {fn} be a sequence of functions converging to / on A such that JA \fn\pdm < oo and JA \f\pdm < oo, 1 < p < oo. Show that lim / \fn\pdm= [ \f\pdm n~>0° J A J A if and only if lim / \fn-f\pdm = 0. 2.3.17. Suppose that {fn} is a sequence of measurable functions on A such that \fn\ < g, where g is integrable on A. Show that / 1™ fndm < lim / fndm < lim / fndm < / lim fndm. J A n—oo n->oo J A n~>oc J A J A n_>0° 2.3.18. Let fn(x) = nxn~l - (n + l)xn, x G (0,1). Show that /. OO OO /. / Y^ fndm ^JZ fndm «y(05l)n=l n=l«/(°'1) and E /mi) \fn\dm = oo. 71=1 2.3.19. Let {fn} be a sequence of measurable functions on A such oo oc that ^2 JA | fn\dm < oo. Show that Y2 fn is integrable on A and 71=1 OO OO / y~" fndm = V / fndm. J*n=l n=lJ*  76 Problems. 2: The Lebesgue Integral 2.3.20. We say that functions /n, n = 1, 2,..., integrable on A are equi-integrable if for every s > 0 there is J > 0 such that for every measurable subset B of A, JB \fn\dm < e for n = 1,2,... whenever m(B) < J. Show that if {/n} is a convergent sequence of equi- integrable functions on a set A of finite measure, then lim / fndm = / lim fndm. n~>°° J A J A n~>°° /A J A1 2.3.21. Prove the following version of Lebesgue's dominated convergence theorem. Suppose {fn} is a sequence of measurable functions converging in measure on A to /. If there exists a function g integrable on A and such that \fn(x)\ < g(x), n = 1, 2,..., x G A, then n—>oo lim / fndm = / fdm. IA J A 2.3.22. Show that the theorem stated in 2.3.20 remains true when convergence is replaced by convergence in measure. 2.3.23. Suppose the sequence {fn} converges in measure to / on a set A of finite measure, and \fn(x)\ < C for x G A, n = 1,2,.... Show that if g is continuous on [—C, C], then lim / g(fn)dm= / g(f)dm. n^°° J A J A 2.3.24. Suppose that {fn} is a sequence of functions defined on a set A of finite measure that converges in measure on A to /. Show that lim / sin(fn)dm = / s'm(f)dm. n~>0° J A J A 2.3.25. Suppose / G Lp[a,6], 1 < p < oo. Show that, given s > 0, there is (i) a simple function </? such that J, bAf — tp\pdm < e, (ii) a step function i/j such that J, bAf — i>\pdm < e. 2.3.26. Find a measurable function / bounded on [a, b] and such that ||/ - ^!|oo = sup{j/(x) - tp{x)\ : x G [a,b}} > 1/2 for all step functions ip.  2.3. Lebesgue Integration 77 2.3.27. Suppose / G Lp[a, b], 1 < p < oo. Show that, given £ > 0, there is a continuous function g such that T 61 |/ — g\pdm < e. 2.3.28. Show, by example, that the result in the previous problem is false if p = oo. 2.3.29. Suppose that g satisfies a Lipschitz condition on R. Prove that if {fn} is a sequence of measurable functions on [a, b] that converges in measure to / and there is a Lebesgue integrable function G such that \fn(x)\ < G(x), then lim / g(fn)dm= / g(f)dm. l^°°J\a,b} J\a,b] >M J[a,b] 2.3.30. Suppose that 1 < p < oo, and that / is measurable on A and such that \f\p is integrable on A. Prove that, given e > 0, there is a continuous function g vanishing outside a finite interval and such that J \f-g\pdm<s. 2.3.31. Let {fn} be a sequence of functions in Lp[a,b], 1 < p < oo, which converge almost everywhere on [a, b] to /. Suppose that tliere is a constant C such that ||/n||p < C for n = 1,2, Prove that for each g in Lq[a, b], l/p + l/q = 1, lim / fngdm = / fgdm. ^°° J[a,b] J[a,b] 2.3.32. Let {/n} be a sequence of functions in Lp[a, b], 1 < p < oo, which converges in norm to / G Lp[a, 6], and let {gn} be a sequence of measurable functions such that \gn\ < C for n = 1, 2 and gn —> g a.e. Show that fngn —> /p in Lp[a, b]. 2.3.33. Show that if / is an essentially bounded function on [a. 6], then \ i/p \f\pdm) =\\f\\x. (Compare with 1.4.41.)  78 Problems. 2: The Lebesgue Integral 2.3.34. Prove the Jensen inequality for Lebesgue integrals (see also 1.6.29). Suppose that / is Lebesgue integrable on [a,b\. If (p is convex on R, then if / fdm < / (p(f)dm. 2.3.35. Show that if m(A) < oo and 0 < pi < p2 < oo, then LP2(A) C LPl(A), and the inequality ll/lk < ll/IUMA))--- holds for /GP2(A). 2.3.36. Show that if / G LPl(A) n LP2(A), where 0 < pi < p2 < oo, and if p\ < r < p2, then / G Lr(A), and the function (p defined by ip(r) = In \\f\\rr is convex on [pi,p2]- 2.3.37. Prove that if / G Ll[a, b] and / ^ 0 a.e. on [a, 6], then lim f—^— ( \f\pdm) =exp[-^— / \n\f\dm) . 2.3.38. For any t G R, the translate of f by t is defined by setting ft(x) = f(x + t). Let / be integrable on R. Show that: (a) JR fdm = JR ftdm. (b) If g is a bounded measurable function, then Km [\g(f-ft)\dm = 0. 2.3.39. Suppose that 1 < p < oo and / G LP(R); that is, JR |/|pdm < oo. Prove that lim JR \f — ft\pdm = 0, where ft is the translate of / by t defined in the previous problem. 2.3.40. Suppose that / is nonnegative and measurable on a set A such that 0 < m(A) < oo. Suppose also that there are positive a and (3 such that ——— / fdm > a and —;—- / f2dm < 6. m(A)V - m(A)JAJ  2.4. Absolute Continuity, Differentiation... 79 Show that if 0 < S < 1 and As = {x e A : f(x) > aS}, then 0 2 m(A(5)>m(A)(l-^)2a 2.4. Absolute Continuity, Differentiation and Integration A real-valued function / defined on [a, b] is said to be absolutely continuous on [a, b] if, given £ > 0, there is 8 > 0 such that J2\fM-f(x'k)\<e k=l for every finite collection {(#&, xj,)} of pairwise disjoint open intervals of [a, 6] with n J2(xk - a*) < ^. fc=i From now on we will deal only with Lebesgue integrals, and we will write Ja f(x)dx instead of f, „ fdm. We will use the following well- known theorems. Theorem 1. Let f be an increasing real-valued function on [a, b\. Then f is differentiate almost everywhere. The derivative f is measurable, and I f'(x)dx<f(b)-f(a). J a Theorem 2. A function f on [a, b] has the form f{x) = /(a) + / g(t)dt J a for some integrable function g on [a, b] if and only if f is absolutely continuous on [a, b]. In this case we have g(x) = f'{x) a.e. on [a, b]. 2.4.1. Prove that if / is absolutely continuous on [a, b], then it is of bounded variation on [a, b]. 2.4.2. Give an example of a function continuous on [a, b] but not absolutely continuous on that interval.  80 Problems. 2: The Lebesgue Integral 2.4.3. Let f(x) = ixas[n^ if ze(<u], [X) [0 if x = 0. Show that if 0 < j3 < a, then / is absolutely continuous on [0,1], but if 0 < a < /?, then it is not absolutely continuous. 2.4.4. Let Jx2|sin(l/x)| if xe(0,l], J \o if i = 0, g(x) = ^/S. Show that / and g are absolutely continuous on [0,1], and that the composite f(g) is absolutely continuous but g(f) is not. 2.4.5. Show that if / and g wee absolutely continuous and g is monotone, then f(g) is absolutely continuous. 2.4.6. Show that an absolutely continuous function / : [a, b] —» R transforms (a) sets of measure zero into sets of me

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