AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS VOL AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS VOL 20 20 The Generalized Riemann Integral Robert M. McLeod The Generalized Riemann Integral The Generalized The Generalized Riemann Integral is addressed to persons who already have an acquaintance with integrals they wish to extend and to the teachers of generations of students to come. The organization of the Riemann Integral work will make it possible for the rst group to extract the principal results without struggling through technical details which they may nd formidable or extraneous to their purposes. The technical level starts low at the opening of each chapter. Thus, readers may follow Robert M. McLeod each chapter as far as they wish and then skip to the beginning of the next. To readers who do wish to see all the details of the arguments, they are given.
The generalized Riemann integral can be used to bring the full power of the integral within the reach of many who, up to now, haven't gotten a glimpse of such results as monotone and dominated conver- gence theorems. As its name hints, the generalized Riemann integral is de ned in terms of Riemann sums. The path from the de nition to theorems exhibiting the full power of the integral is direct and short. M A PRESS / MAA AMS
4-color Process 290 pages spine: 9/16" finish size: 5.5" X 8.5" 50 lb stock 10.1090/car/020
THE GENERALIZED RIEMANN INTEGRAL
By
ROBERT M. McLEOD THE CARUS MATHEMATICAL MONOGRAPHS Published by THE MATHEMATICAL ASSOCIATION OF AMERICA
Committee on Publications E. F. BECKENBACH, Chairman
Subcommittee on Carta Monographs DANIEL T. FINKBEINER, Chairman RALPH P. BOAS DAN E. CHRISTIE ROBERT GILMER GEORGE PIRANIAN HE CARUS MATHEMATICAL MONOGRAPHS are an ex- pression of the desire of Mrs. Mary Hegeler Carus, and of her son, Dr. Edward H. Carus, to contribute to the dissemination of mathematical knowledge by making accessible at nominal cost a series of expository presentations of the best thoughts and keenest researches in pure and applied mathematics. The publication of the first four of these monographs was made possible by a notable gift to the Mathemati- cal Association of America by Mrs. Carus as sole trustee of the Edward C. Hegeler Trust Fund. The sales from these have resulted in the Carus Monograph Fund, and the Mathematical Association has used this as a revolving book fund to publish the succeeding monographs. The expositions of mathematical subjects which the monographs contain are set forth in a manner comprehensible not only to teachers and students specializing in mathematics, but also to scientific workers in other fields, and especially to the wide circle of thoughtful people who, having a moderate acquaintance with elementary mathematics, wish to extend their knowledge without prolonged and critical study of the mathematical journals and treatises. The scope of this series includes also historical and biographical monographs.
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No. 20. The Generalized Riemann Integral, by ROBERT M. McLEOD The Cams Mathematical Monographs
NUMBER TWENTY
THE GENERALIZED RIEMANN INTEGRAL
By
ROBERT M. McLEOD Kenyan College
Published and Distributed by THE MATHEMATICAL ASSOCIATION OF AMERICA © 1980 by The Mathematical Association of America (Incorporated)
Library of Congress Catalog Card Number 80-81043
Hardcover (out of print) ISBN 978-0-88385-021-3 Paperback ISBN 978-0-88385-045-9 eISBN 978-1-61444-020-8
Printed in the United States of America
Current printing (last digit): 10 9 8 7 6 5 4 3 2 1 PREFACE
In calculus courses we learn what integrals are and how to use them to compute areas, volumes, work and other quantities which are useful and interesting. The calculus sequence, and frequently the whole of the undergraduate mathematics program, does not reach the most powerful theorems of integration theory. I believe that the general- ized Riemann integral can be used to bring the full power of the integral within the reach of many who, up to now, get no glimpse of such results as monotone and domi- nated convergence theorems. As its name hints, the gener- alized Riemann integral is defined in terms of Riemann sums. It reaches a higher level of generality because a more general limit process is applied to the Riemann sums than the one familiar from calculus. This limit process is, all the same, a natural one which can be introduced through the problem of approximating the area under a function graph by sums of areas of rectangles. The path from the definition to theorems exhibiting the full power of the integral is direct and short. I address myself in this book to persons who already have an acquaintance with integrals which they wish to extend and to the teachers of generations of students to come. To the first of these groups, I express the hope that the organization of the work will make it possible for you to extract the principal results without struggling through technical details which you find formidable or extraneous to your purposes. The technical level starts low at the
vii viii PREFACE opening of each chapter. Thus you are invited to follow each chapter as far as you wish and then to skip to the beginning of the next. To readers who do wish to see all the details of the arguments, let me say that they are given. It was a virtual necessity to include them. There are no works to refer you to which are generally available and compatible with this one in approach to integration. I first learned of the generalized Riemann integral from the pioneering work of Ralph Henstock. I am in his debt for the formulation of the basic concept and for many important methods of proof. Nevertheless, my presenta- tion of the subject differs considerably from his. In partic- ular, I chose to use only a part of his technical vocabulary and to supplement the part I selected with terms from E. J. McShane and other terms of my own devising. I wish to express my appreciation to the members of the Subcommittee on Carus Monographs for their encourage- ment. I am particularly indebted to D. T. Finkbeiner. His support enabled me to persevere through the years since this writing project began. He and Helene Shapiro also worked through an earlier version of the book and pro- vided helpful comments. I thank the Department of Mathematical Sciences of New Mexico State University for its generous hospitality during a sabbatical leave year devoted in large part to the writing of the first version of the book. Finally, I thank Jackie Hancock, Joy Krog, and Hope Weir for expert typing.
Gambier, Ohio ROBERT M. MCLEOD January, 1980 LIST OF SYMBOLS
PAOB 7 gauge 10 R real numbers 17 R» ^-dimensional Euclidean space 17 Euclidean length 17 divisions, tagged divisions 17 zJ tagged interval 18 HJ) length of interval J 18 fL(zJ) term in Riemann sum 18 /L(q>) Riemann sum 18 integral on [a, b] 19 R extended real numbers 22 °°D length of unbounded interval 23 R> extended /^-dimensional space 31 / closed interval 31 M(/) measure of interval / 33 /Λί(Φ) Riemann sum 33 /// integral on interval / 33 hi integral on a set Ε 34 Í extended positive integers 36 increment of F 43 lim/M(<2>) limit notation for the integral 73
sum of integrals 77
ix ÷ LIST OF SYMBOLS
sum of absolute values of integrals 77 supremum of / and g 82 /A* infimum of / and g 82 / + positive part of / 82 r negative part of / 82 lim/; lim/(x) limit according to a direction 94 S x, s lim | lim fix, y)\ iterated limits 94
/M-null null set 106 XE characteristic function of Ε 123 measure of Ε 123 /"'(£) inverse image 133 /(·./) the function χ -* f(x, y) 152 Φ-null null set 159 /Δα(<3) sum for Riemann-Stieltjes integral 177 (Λ)/· /rfa norm integral 177 refinement integral 177 (β)Λ /<*» gauge integral 177 11*11 norm of a division 181 9t-limit norm limit 181
Saf(x,y)dx line integral 216 (£)/// Lebesgue integral 235 CONTENTS
PAGE PREFACE vii LIST OF SYMBOLS ix INTRODUCTION I CHAPTER 1—DEFINITION OF THE GENERALIZED RIEMANN INTEGRAL 5 1.1 Selecting Riemann sums 7 1.2 Definition of the generalized Riemann in- tegral 17 1.3 Integration over unbounded intervals 21 1.4 The fundamental theorem of calculus 25 1.5 The status of improper integrals 28 1.6 Multiple integrals 31 1.7 Sum of a series viewed as an integral 36 51.8 The limit based on gauges 38 51.9 Proof of the fundamental theorem 40 1.10 Exercises 45 CHAPTER 2—BASIC PROPERTIES OF THE INTEGRAL 47 2.1 The integral as a function of the integrand 48 2.2 The Cauchy criterion 50 2.3 Integrability on subintervals 51 2.4 The additivity of integrals 52 2.5 Finite additivity of functions of intervals 55 2.6 Continuity of integrals. Existence of primitives 58 2.7 Change of variables in integrals on inter- vals in R 59 S2.8 Limits of integrals over expanding intervals 65 2.9 Exercises 68 xi xii CONTENTS
CHAPTER 3—ABSOLUTE INTEGRABILITY AND CON- VERGENCE THEOREMS 71 3.1 Henstock's lemma 74 32 Integrability of the absolute value of an integrable function 77 3.3 Lattice operations on integrable functions 81 3.4 Uniformly convergent sequences of func- tions 83 3.5 The monotone convergence theorem 86 3.6 The dominated convergence theorem 88 53.7 Proof of Henstock's lemma 90 53.8 Proof of the criterion for integrability of l/l 91 53.9 Iterated limits 93 53.10 Proof of the monotone and dominated convergence theorems 96 3.11 Exercises 101 CHAPTER 4—INTEGRATION ON SUBSETS OF INTERVALS 103 4.1 Null functions and null sets 104 4.2 Convergence almost everywhere 110 4.3 Integration over sets which are not inter- vals 113 4.4 Integration of continuous functions on closed, bounded sets 116 4.5 Integrals on sequences of sets 118 4.6 Length, area, volume, and measure 122 4.7 Exercises 128 CHAPTER 5—MEASURABLE FUNCTIONS 131 5.1 Measurable functions 133 5.2 Measurability and absolute integrability 135 5.3 Operations on measurable functions 139 5.4 Integrability of products 141 S5.5 Approximation by step functions 143 5.6 Exercises 147 CHAPTER 6—MULTIPLE AND ITERATED INTEGRALS 149 6.1 Fubini's theorem 150 6.2 Determining integrability from iterated integrals 154 CONTENTS xiii
56.3 Compound divisions. Compatibility theorem 164 56.4 Proof of Fubini's theorem 168 56.5 Double series 171 6.6 Exercises 173 CHAPTER 7—INTEGRALS OF STIELTJES TYPE 177 7.1 Three versions of the Riemann-Stieltjes integral 180 7.2 Basic properties of Riemann-Stieltjes integrals 183 7.3 Limits, continuity, and differentiability of integrals 187 7.4 Values of certain integrals 189 7.5 Existence theorems for Riemann-Stieltjes integrals 192 7.6 Integration by parts 195 7.7 Integration of absolute values. Lattice operations 200 7.8 Monotone and dominated convergence 204 7.9 Change of variables 205 7.10 Mean value theorems for integrals 209 57.11 Sequences of integrators 211 57.12 Line integrals 214 57.13 Functions of bounded variation and reg- ulated functions 220 57.14 Proof of the absolute integrability theorem 225 7.15 Exercises 227 CHAPTER 8—COMPARISON OF INTEGRALS 231 58.1 Characterization of measurable sets 232 58.2 Lebesgue measure and integral 234 58.3 Characterization of absolute integrability using Riemann sums 236 8.4 Suggestions for further study 244
REFERENCES 245 APPENDIX Solutions of In-text Exercises 247 INDEX 269
246 COMPARISON OF INTEGRALS Ch.8
8. , A unified theory of integration, Amer. Math. Monthly, 80 (1973) 349-359. 9. E. J. McShane and Truman Botts, Real Analysis, Van Nostrand, Princeton, 1959. 10. Μ. E. Munroe, Measure and Integration, 2nd ed., Addison- Wesley, Reading, Mass., 1971. 11. A. E. Taylor, General Theory of Functions and Integration, Blaisdell, Waltham, Mass., 1965. 12. J. H. Williamson, Lebesgue Integration, Holt, Rinehart and Winston, New York, 1962. APPENDIX
SOLUTIONS OF IN-TEXT EXERCISES
Chapter 1
1. Begin by choosing an integer m so that l/m < c. Enclose each number 1// in an interval [l/j — dj, l/j + dj] which is contained in y(l/j") for j = 1, 2,..., m. These can also be chosen so that l/j + d., < l/(J1) - d}_, for j = 2,3,...,m and 0 < l/m - dm. Tag [l/j - dj, l/j + dj] with l/j for 2 [1 - dx, 1] with 1. The rest of [0, 1] is [0, l/m - dm] and each of the intervals [l/j + dj, l/(j - 1) - dj_x] for 2 < j < m. Tag the former with 0 and the latter with any of its points, say l/j + dj. Now we have a division of [0, 1] with 2m intervals tagged in such a way as to be γ-fine. 2. (a) Let 35 e Rs and let zJ Ε Φ. Since / has length less than δ it is contained in any open interval of length 28 centered on a point of J. Thus / C (ζ — δ, ζ + δ). In consequence Φ £ GRa. When Φ ε GRg and ζ/Ε^/ςίζ-ί,Η δ). Since the length of (ζ - δ, ζ + δ) is 2δ and the endpoints of J are between ζ — δ and ζ + δ, L(J) < 28. Consequently Φ ε R2S. (b) Suppose the Riemann integral of / on [a, b] exists. b Choose δ so that \J af-fL(^)\ < e when L(J) < δ for all 247 248 APPENDIX zJ in That is, the inequality holds for all <3) in RB. Let γ(ζ) = (ζ - δ/2,2 + δ/2) for all ζ in [a, b]. Then the set of all γ-fine is GRs/2. Since GRs/2 Q Rs, the gener- alized Riemann integral of / exists and is the same as its Riemann integral. Now let/be a generalized Riemann integrable function with the added property that fL^)\ < £ for a11 60 in GRS for some positive δ. Since Rs C GRS the Riemann integral of / also exists. Now we have characterized the Riemann integrable functions among the generalized Riemann integrable functions as those for which there is a gauge γ(ζ) = (ζ — δ, ζ + δ) with constant δ such that |/*/-/i(^)l < £ for all γ-fine Φ. 3. Let f(x) = 0 when χ is irrational. Let p/q be a fraction in lowest terms. Set f(p/q) = q- Since the rationals are countable this function is integrable on any interval [a, b] and its integral is zero, according to Example 4, p. 19. Also / is unbounded on every interval [c, d] for the following reasons. Let Μ be given. Select a prime number q such that q> Μ and 2/q d that χ < d, i xf= F(d) - F(x) and /*/= G(d) - G(x). Thus F(x) = G{x) + Κ where K= F(d)- G(d) when x χ e / with χ < d. When xEl &ndd Since / is also integrable on [s, b], there is a gauge γ, such that |/f/ — /£(£)! < c when S is a γ,-fine division of [s, b\. It is possible to choose ys so that ys(z) C y(z), too. Choose c so that c e y(a) and |/(a)|L([a, c]) < c. Let s £ (a, c). Let S be a γ,-fine division of [s, b]. Let Φ = {a[a, j]}US. Then ^ is a γ-fine division of [a, b]. Now 6 Pf-ffUlff-M *) + /M6)-fV| + \f(a)L([a,s])\. Each term on the right is less than «. Hence 6. We know that Jgf exists if and only if Ιύη,^/ό/ exists. Also /"/ = lim^^/o/. Using a primitive of / on [0, /] we get J0/= + (* - «Κ,+ι where w < / < m + 1. When /"/ exists we specialize t to integer values and get Jo7= »X.-.«Jo/= Σ"-ι«*· Conversely, when the series converges, limm^00am+I = 0. Thus lim^^/o/ -1™*-..οΣ~-ι«*βΣ"-α· 7. (a) This is the scale: I I I I I I I I I ' III -00-9-4 -2 -1 -1/2 0 1/2 1 2 4 9 oo (b) Clearly h is strictly increasing and maps [—oo, oo] onto [-1, 1]. Given χ and y in with x¥=y, there is some coordinate where they differ, say x^y,. Then λ(χ,) φ h(yt) so that H(x) φ H(y). We have shown Η is one-to-one. To show that it is onto, take y_ such that — 1 < < 1 for 1 < i < p. There is xt in R such that Ηχί)-)Ί· Hence Η maps the point χ with these components xt onloy. 250 APPENDIX Since h is increasing it maps [u, v] onto [«(«), h(v)]. Consequently Η maps [κ,, ©,] X — X [up, vp] onto the Cartesian product of the intervals [«(«,), h(v,)]. The same is true of intervals other than closed intervals. (c) Draw a square. Put on its edges the scale shown in part (a). Chapter 2 G 1. Fix γ, and γ2 so that \fM( b) - f,f\ < c when is γ,-fine and | g3/(<>D) - j,g\ < c when Φ is γ2-ίίηβ. Let γ(ζ) C γ,(ζ) Π γ2(ζ) for all ζ in [a, b\. Then Φ is γ,-fine and γ2-ίΐηβ whenever it is γ-fine. Thus (/+g)M(^)-J//-J/ when <Φ is γ-fine since (/+ g)M(^) = /Μ(<Φ) + gM(<3)). This shows that /+ g is integrable and that ///+ //g is its integral. Since (ς/)Μ(<Φ) = c(fM(q))) we also have (c/)M(fiD) - c£/| < |c||/M(*D) - J^/| < |c|c when ^ is γ-fine. Thus c/,/ is the integral of cf. A standard induction argument shows that 2A-IC*/A *S c integrable when each fk is integrable and that 2*-i *///* is its integral. 2. Let γ be a gauge on / such that \$if- }Μ(0))\ < e and \fjg — gM^)] < c when <Φ is γ-fine. Since g is real valued, gM(<%) < J,g + e. Also |/M(<>D)| < gM^) for all SOLUTIONS FOR CHAPTER 2 251 <>D since |/| < g and Μ takes nonnegative values. Thus Since e is arbitrary, \fjf\ < ftg, as claimed. 3. By hypothesis, for each whole number η there is a gauge γ„ such that l/Mf^D) -/M(S)| < 1/u when and S are γ,,-fine divisions of /. We may also suppose 1j(z)Ql,(z) when / γ2(ζ) η · · · Π γ„(ζ).) For each Λ fix a γ,,-fine division ^)Λ, |/il/(^.)-/A/(^.)| It follows easily that (/Λ/ί^η))™»! is a Cauchy sequence in R?. Consequently it converges to a limit A in R?. To show that A is the integral of /, fix Ν so that 1/JV < e/2 and \A -fM(<$N)\ < e/2. Let <Φ be γΛ,-fine. Then |/M(*D) - A\ < |/Μ(Φ) -fM(%)\ + |/M(%) - Λ| < l/N + €/2<€. ThmA = i,f. 4. The techniques needed for the solution of this exercise anticipate the discussion on p. 75. There it is shown that & is a subset of a division "D of I. Let 9 = - g. Then v(&) + v(<5) = vi6!)). Since 0 < v(J) for all J, 0 < v(9). Moreover i>(fy) = y(7) since ν is finitely additive. Consequently r(&) < ?(/). 252 APPENDIX 5. Let Ε, = {χ £ [a, ft]: a(x) = t). Then £,CC when ι = oo and when t = — oo. When t is finite we need to show that E, — C is countable. Let χ e E, — C. Then σ\χ) Φ 0, consequently there is δχ such that σ(γ) φ σ(χ) when 0 < \γ — x\ < δχ. In other words, the set E, — C is made up of isolated points. It must be countable. To see this, let F„ = [χ £ [-η,«] : δχ > 1 /η}. Then £, - C = LC-i^ir ^ suffices to show that each Fn is finite. Form a division of [—Μ, /i] into subintervals of length l/n. Each subinterval contains at most one point of Fn. Thus Fn is finite. 6. There are gauges γ, and y2 on [a, c] and \d, b] such that \iCaf-fL(^>i)\ < e/2 wh«n <φ, is a γ,-fine division of [a,c] and /L(<5>a)| < c/2 when % is a γ2-Ηηβ division of [d, b]. We may also suppose γ,(ζ) C γ(ζ) for all ζ in [a, c] and γ2(ζ) C γ(ζ) for all ζ in [d, ft]. Fix 3), and <ΐ)2. Let S be a γ-fine division of [c, d]. Set Φ = Φ, U S U %. Then Φ is a γ-fine division of [a, ft]. Consequently, |£7 - /L(S)| < |f/ - /L( This uses the finite additivity of the integral. Now these terms are less than «, c/2, and e/2. Hence the conclusion follows. (If a = c or ft = d, some terms become zero but otherwise the argument goes the same.) Note: A sharpening of the argument above shows that \iif- fL(&)\ < c. This latter inequality is a special case of Henstock's lemma. (See p. 74.) 7. (a) When / > 0 the function s-»/*/ is decreasing SOLUTIONS FOR CHAPTER 3 253 on the interval (a, b) and nonnegative. Such a function is bounded above if and only if linv^J* / exists. Existence of this limit is exactly the criterion for existence of /*/. (b) The function F : ί -»/* / has a limit at a provided the Cauchy criterion is satisfied. That is, it suffices to find c such that \F(t) - F(s)\ < e when a < s < t < c. Since J*g exists, lim^/jg does exist. Hence there is c such that Msg ~ Jfel < £ wnen a |F(i)-F(5)| = f'f Chapter 3 1. Since Ιϊτα,^^Ι/λ[χ dx = oo, we know that Jf°l/Vx dx does not exist. Set fn(x)= \/Jx when 1 < χ < η and f„(x) = 0 when n< x. Then fn converges uniformly to 1/^Jc" on [1, oo). 2. The function / is integrable on [a, b]. Set F(c) = lim^^F^c) and F(x) = F(c) + Jxf elsewhere in [a, b\. (We are using the oriented integral here.) We are going to get an inequality from which two separate deductions can be made. By hypothesis there is nt such that )/„(*)-/0c)| < c for all χ in [a, b] and all η > nt. Let u and υ be in [a, b]. Then \W - Fn(u) - (F(v) - F(u))\ = |f (/„ -/) < e\v — u\ when η > nf. The first conclusion is uniform convergence of Fn to F. To get it let u = c. Then one more application of the 254 APPENDIX triangle inequality yields \F„(v) - F(v)\ < \F„(c) - F(c)| + e{b - a). Uniform convergence follows easily. The next conclusion is uniform convergence of difference quotients. Let u = χ and υ = χ + t with t φ 0. Divide by \t\. Then Fn(x + t)-Fn(x) F(x + t)-F(x) < € for all χ and t such that χ and x + r are in [a, b] and all « > «e. Choose χ so that F„'(x) = /,(x) for every integer n. Select 8. such that Fn(x + t)-Fn(x) when η = «t and |*| < St. Now F(x + r) - F(x) F(x+0-F(x) Fn(x + r)-Fn(x) F„(x + t)-F„(x) + !/„(*)-/0)Ι· Each term is less than c when η > nt and \t\ < 8f. Consequently F'(x) = /(x). Let C„ be the set of values of χ for which we cannot claim that F„'(x) = /„(x). Then C„ is countable and (J"_,Cn is also countable. Outside this set we know that SOLUTIONS FOR CHAPTER 4 255 F' = /. Since F is continuous, it is a primitive of / on [a, b]. (This argument can be greatly abbreviated by use of the iterated limits theorem given in Section S3.9.) 3. It is enough to consider increasing functions. We may apply the proposition proved in the preceding exercise. Thus it is our goal to define functions f„ which converge uniformly to / and have primitives. Step functions are the right choice. Since / is increasing, for any number Κ in [f(a),f(b)] there is at least one number c in [a, b] such that f(x) < Κ when χ < c and Κ < f(x) when c < x. Let the integer η be given and set 4, = (/(&)-/(«))/«. For ι = 1, 2,..., η - 1, let Kt = /(a) + id„. Choose a corresponding c,. Set c0 = a and c„ = b. Define/, so that/„(c,) = /(c,) for ι = 0,..., η and fn(x) = Kj when c,_,< *< \f(x)-f„(x)\ < dn for all χ in [a,b]. Consequently /„ converges uniformly to / on [a, b\. Chapter 4 1. Recall that Ε is g/Ai-null whenever Ε is /A/-null. Now apply this with the constant function with value 1 in place of / and/ in place of g. Thus Ε is/M-null when Ε is M-null. By definition this means that |/|M(S) < c for every γ-fine partial division S whose tags are in E. When Φ is a division of / and & is the subset of Φ whose tags are in E, I/IMC^) = |/|M(S) since / vanishes outside E. Thus Γ,|/| = 0. 2. There is a gauge γ on I such that I/IM^5))) < c when Φ is a γ-fine division of /. Let Ε = {χ e / : f(x) φ 0}. Let S be a γ-fine partial division of / whose tags are in E. There is a γ-fine division <Φ such that & C <>D. Then 256 APPENDIX |/|M(S) = |/|Λ/(<3))<€. Thus £ is |/|M-null. Let g(x) = l/l/OOl when χ EE. Then £ is g|/|M-null, i.e., M-null. 3. Let M' and Μ be the interval measures in Rp~1 and IV, respectively. For a given c we must define γ on £ so that M(S) < c when S is a γ-fine partial division with tags in £. Using the continuity of / choose γ,(ζ) so that |/(x)-/(z)| < e' when χ Ε / Π γ,(ζ). Fix a γ,-fine division Φ, of /. For each zJ in 3), let //, be the interior of 7, i.e., the open interval obtained by peeling the faces from /. Let Β be the union of all faces of all J of 3),. Then Β X R is an M-null subset of Rp since it is a finite union of degenerate intervals. Thus there is γ defined on Β X R such that M(S) Set γ(χ,/(χ))=///Χ(/(ζ)-€',/(ζ) + €')· Let S be γ-fine with tags in £. Separate S into subsets S, and S2 having tags in Β X R and (I-B)XR. Then M(S,) < c/2 and M^ < 2yW'(7)2c' < 2c'M'(/). (The first inequality on M^&j) results from grouping together all terms associated with a single J of 3),.) The choice e' = c/(4M'(/)) gives M(S) < c, as desired. 4. According to Example 5, p. 86, the functions g^(i~lf„) are integrable. There is a bound A for /,/,. Since g Λ (/-'/„) < Γ% for all «, Λ (Γ%)) < i~lA. Since/, < fn+, the sequence g Λ (/ '/„) increases for fixed i. Thus JjA, = lmv^JVtg Λ (r'/„)) < ΓιΑ, too. To see that lim^^A, = A, first let χ ε £. Then g Λ (i Ά„) has value 1 at χ for all large n. Hence A,(x) = 1 and ϋπι,.^Α,ΟΟ = 1. When xEl-E, (gA(r%))(x) < 1 '/(x), hence A,(x) < i '/(x). Consequently, hm^^h^x) = 0. A second use of monotone convergence gives f,h = lim^JjA, = 0. SOLUTIONS FOR CHAPTER 4 257 5. The essentials of the solution can be conveyed by an example in N. Consider the sequence 1, — 1, 1, 1/2, — 1/2, 1/2, 1/3, —1/3, 1/3,... . This is our function •f: Ν -» R. Note that the terms go in groups of three. Let's replace every third member by zero. That means integrate / over Ε = {1, 2,4, 5, 7, 8,... }. It is easy to see that exists. Next replace the first member of each group by zero; i.e., integrate / over F= (2, 3, 5, 6, 8, 9,... }. Integration over Ε Π F is the same as summing the middle terms of each group. These terms are negatives of terms of the harmonic series. Thus JEnF f does not exist. Neither does fEUF f for much the same reason. 6. For each positive integer n, partition / into a finite collection % of pairwise disjoint intervals Κ such that the diameter of Κ is no more than l/n. Also make the choices so that each interval of 3FB+, is contained in an interval of <5n; i.e., &n+l is a refinement of 9n. Define fn from <&„ as follows. For each χ in /, there is just one interval Κ of 9„ to which χ belongs. Let f„(x) = lub{/(x) : xE K). This is meaningful everywhere since / is bounded above. (Recall that / has a maximum on Ε and / is constant on I — E.) The sequence (/,)*_, decreases since 9n+l is a refinement of 9„. Now let χ Ε I — E. There is an open interval y(x) which does not intersect Ε since Ε is closed. Every interval of sufficiently small diameter containing χ is a subset of y(x). Since / vanishes on y(x) so does /„(*) = 0 for sufficiently large n. When χ ε Ε there is an open interval y(x) such that l/OO -/(x)| For large enough η the interval of 9n which contains χ is a subset of γ(χ). Thus/,(χ) < /(χ) + e for all sufficiently 258 APPENDIX large π. The definition of /„ also insures that/(x) < fn(x\ Hence/(x) < f„(x) < f(x) + c for sufficiently large n. We have shown that /(x) = limfl^00/I(x) for all χ Ε I, as required. 7. Set Ει = Hl and E„ = Hn- for η > 2. Note that Hn_lQHn. Thus/ is integrable on En. Since /is also bounded on each E„, it is absolutely integrable on E„. Moreover JE \f\ < bn(M(H„) - M(//„_,)) for η > 2. But M(Hn) - M\Hn_x) = (2ny - (2(« - 1))' < pVn'~'. Thus ΣΪ-Ι/Α,Ι/Ι is convergent since Σ?-ιΜ'_1 is assumed convergent. This implies absolute integrability of/. 8. We know that |/| is integrable on En for all η since l/l < g and g is integrable on /. By induction / is absolutely integrable on Ui-i^i f°r au* n since absolute integrability carries over to unions of two sets. Let F, = £, and Fn = U7-i^i" (U"-/^) when Λ > 2. Now/ is absolutely integrable on each member of the pairwise r disjoint sequence (£„)"_ P Moreover 27-iJV„l/l < //£ f° all n. Thus ΣΠ-IJFJ/I is finite- % tne countable additivity proposition / is absolutely integrable on IX-,F„, i.e.,on£. 9. The characteristic functions of Π7-ι^ί decrease to the characteristic function of Πί^ι-^ί·· Thus the monotone convergence theorem yields the integrability of the intersection of the sequence. Now U7-i^i are integrable sets whose characteristic functions increase to the characteristic function of the union of the sequence. Since MflJi-i^/) < Σ <-iand MiUi-i^i) < M(^) when US-i^, is contained in the integrable set F, the monotone convergence theorem implies integrability of \JT,iEf under each of the conditions which have been given. 10. The conclusions in (a) follow from known properties of integrable sets, since the intersection with a SOLUTIONS FOR CHAPTER 5 259 bounded interval / distributes over the other operations. For instance, (E U F) Π J = (Ε Π /) U (F Π J). Thus (£Uf)ni is the union of integrable sets and is integrable itself. The others go similarly. Measurability of ΠΓ-ι^/ι an<* U^-i^n follow similarly. The inequality is trivial when the right-hand side equals oo. When it is finite, every set En is integrable and we know from the previous exercise that U^-i^n *s mte" grable. Moreover u(|J?» < Σ7-1 < ΣΓ-1 μ(3) for all n.Monotone convergence tells us that p(\JT-\E,) When the sets are nonoverlapping and Uif-i^n is not integrable the inequality just proved becomes equality. When UJf-i^n *s integrable, we are back to countable additivity of integrable sets since every E„ must also be integrable. Chapter 5 1. We prove that (i) implies (ii) implies (in) implies (iv) implies (i). Assume (i). Let G = (-oo, a] and G' = (a, oo). Then f~l(G) = /-/_1(G'). Since /"'( Assume (ii). Let G = (- oo, a) and set Gn = (— oo, a — 1 l/n]. Then \J^Gm = G and/"'(G) = υϊ-ιΓ ^)· By (ii) each / \G„) is measurable. Thus / (G) is also measurable and (iii) holds. Prove (iv) from (iii) like (ii) from (i). Deduce (i) from (iv) on the model of (iii) from (ii). Measurability of / obviously implies (i), hence all the others. Assume (i) through (iv). Then f~l(G) is measurable for every unbounded open interval. Since 260 APPENDIX (α, b) = (α, oo) - [b, oo) and /"'((a, b)) =/"'((«, oo)) - /_1([6, oo)), it follows from (i) and (ii) that /"'(G) is measurable for bounded open intervals, too. Thus / is measurable. 2. In R* let J„ = [- η, η) X · · · X [-n, n). Partition each factor [-n, n) into n2" intervals of length 1/2"-1. Let each subinterval be closed on the left and open on the right. Form a partition 9„ of J„ by taking Cartesian products. (See Fig. 3, p. 144.) For each G ef„ the set /„ η f~\G) is a bounded measurable subset of /. Let Κ be the closed interval having the same faces as G. Then Κ has on its boundary a unique pointy which is nearest to the origin. Set f„(x) =y for all χ in /„ Π f~\G). Do this for all G in This defines /„ on /„ Π f~\J„). Set /„(x) = 0 elsewhere. Then r x m IΛ Ml ** I/Ml f° ^1 ^ Since 9n+l is a refinement of The convergence of fn(x) to /(x) is proved as follows. Let χ ε / η R'. There is nx such that χ e /„ and/(x) e 7n when η > nx. Then /(x) is in some G of Sj,. Since /„(x) belongs to the closed interval having the same faces as G, 1/00 -/„(*)! <^/2n~1. Consequently, lim^^/^x) =/(x). 1 3. Let £ = U"-.U~-.n*-m/*" «?n)- Let χ EE. Then χ e D?-m/* '(G„) Ior some m and n. Consequent- ly, fk(x) > a + l/n when k > m. Thus lim^^/^x) > a + l/n. From this we see that χ e/~'(G). Thus £ C /-•(ο. Conversely, let xEf '(G). Fix η so that α + l/n < f{x). There is m such that /t(x) > a + l/n when k> m. _ Then χ e/*-'(G) when k> m; i.e., χ e f|?- J* '(Gn). Consequently, χ EE. This completes the proof that /"'(G) C £. Thus the sets are equal. SOLUTIONS FOR CHAPTER 6 261 4. If one of / and g is a null function the same is true of fg. Then both sides of the inequality are zero. Suppose neither / nor g is a null function. Then J J7|/| >0 and J/|g|'>0. We can choose a positive constant c so that jj\cf\s = 1. In fact we need only have i-'-J/l/f or Similarly, f,\kg\'=l when k 1 = (Jj\g\')l/>- Integration on both sides of \(cf)m\<\cfr/s+\kg\'/t yields Hence < c '/c ', as desired. Chapter 6 1. Some notation is needed. Suppose / C R^. Let the /•-fold integration be accomplished by expressing R^ as J>, Χ P2 X · · · X PT with r < p. Write the intervals as products like this: / = /, Χ I2 X · · · X Ir where I} C Pj for 1 < j < r. We may assume 1—GOH. Since G and Η do not overlap, there is an integer k such that Ij = Gj — Hj when j Φ k and GkU Hk = Ik with Gk and i/A nonoverlapping. For convenience denote iterated integrals as follows. When / < r let the result of the first / integrations over an interval J be p(J;xi+1,...,xr) Let v(J) be the result of the r-fold integration. 262 APPENDIX The equalities Ij = Gj = Hj when j < k imply v(I\ Jc/+i, · · ·, xr) = v(G\ xi+l, ...,xr) = v(H; xi+1,..., xr) when ι < k. The additivity of integrals allows us to replace the second equality by summation when / = k. The result is either V G "(J; Xk+u ···,*,) = ( > · · ·. xr) + v(H; xk+i,... ,xr) when k < r or, of course, v(I) = v(G) + if it happens that k = r. When k < r the linearity of integration applies in the last r — k integrations to yield the desired final conclusion that v(I) = v(G) + v(H). The equalities Ij = Gj = Hj for j = k + 1,..., r must be used in these last r — k integrations. 2. Since U can be expressed as a countable union of bounded sets, it is enough to give the solution for a bounded interval /. It is enough to show that there is a gauge γ on U such that |/(ζ)Μ(/)-φ(7)|<€Μ(/) when ζ GJ and / C γ(ζ). When ζ e U, the continuity of / at ζ implies that there is y(z) such that \f(x)- f(z)\ < c when xGl Π γ(ζ). Let / C / Π γ (ζ). On / consider the constant function g such that g(x) = f(z). The iterated integral of g over J has the value /(z)M(/). Then The inner integral exists because, for fixed x, f(x,y) vanishes on [c, g(x)) and (A(x), d] and is continuous when restricted to [g(x), h(x)]. Let [r, s] C [c, d\. The outer integral exists if we show J'rf(x,y)dy is a continuous function of x. Thus it is s s appropriate to consider J rf(x,y) dy — J rf(u,y) dy, i.e., i'[f(x,y)—f(u,y)]dy. It is important to estimate this integrand. Let Kx = [r, s] Π [g(x), A(x)] for each χ in [a, b]. The uniform continuity of/on Ε implies that \f(x,y)-f(u,y)\ < € when \x — u\<8 and y e Kx Π A„. Since / is bounded, there is a constant Λ such that \f(x,y) — f(M,y)\ < A for all x, u, and 7. Next note that f(x,y) — f(u,y) = 0 when/ is outside Kx U K^. Finally, note that the total length of the intervals making up (Kx U^)- (Kx Π Κγ) is no more than | g(x) — g(u)\ + \h(x) — h(u)\. Now suppose |x — u\ < 8. The estimates of the preceding paragraph give < tL(Kx Π K„) + A(\g(x) - g(u)\ + \h(x) - A(«)|). Since L(KX Π KJ < d— c and g and A are continuous, the desired continuity of the inner integral follows. Now on / - (G U H) and this set is (/M - Chapter 7 1. Let A be the 9l-limit of /Δα(3)). Choose δ so that \A - /Δα(<3))| < c when ||Φ|| < δ. Fix a division f with H^ll < δ. Let Φ be a refinement of <5. Then < < δ. Hence \A - /Δα(<Φ)| < c. Therefore A is also the A-limit οί/Δα(<ϊ)). Now assume A is the 91 -limit. To show that A is the S-limit requires the use of a special gauge associated with a division. Fix a division f for which \A —/Δα(<ΐ>)[ < e when 3) is a refinement of 'S. Let γ? be defined so that y9(z) contains no endpoint of 9 distinct from z. Let <>D be Y¥-fine. Then every endpoint of *$ appears as tag of each interval of 3) which contains that endpoint. Form S from <3) by replacing z[u, v] by z[u, z] and z[z, v] when u < ζ < υ. Then S is a refinement of ff. Moreover /Δα(<$) = /Δα(δ). Thus L4 -/Δα(<35)| = μ -/Δα(8)| < c because of the choice of ff. 2. It is enough to find γ for which |/Δα(Φ) -/^Δ^(Φ)| < c for all γ-fine 3). From this statement and the triangle inequality we can deduce existence of both integrals from existence of either of them. Let En = {JC Ε [α, b] : η - Κ |/(JC)| < η). These sets E„ cover [a, b] and are pairwise disjoint. Any division Φ falls into subsets <>D„ having tags in En. Now |/Δα(3)„) - fg^(%)\ < n\Aa - gAfi\(%). Thus it is our goal to η Δα - c wnen define γ so that Σ/ι Ι g^fi\(^K) < ^ is γ-fine. Henstock's lemma gets us to our goal. SOLUTIONS FOR CHAPTER 7 265 Begin with gauges y„ on [a, b\ so that |/*g&j8 - gAP(fy)\ ,+1 Thus |Δα-^|(Φ„)<2α£/(/ισ2' ) and |/Δα(<Φ„)- /^Δ/3(Φη)| < c/2". Summation on η does the rest. 3. The step function / can be expressed as a linear combination of functions of the types considered in Example 3, p. 190, and Example 4, p. 190. The η open intervals give us functions gj such that gXx) = 1 when χ e.(Xj_lt Xj) and gy(jc) = 0 elsewhere. The n + 1 end- points give us functions Ay such that hXxj) = 1 and hj(x) = 0 elsewhere. Then/= Σ"-ι*& + Σ;-ο/(*/)Λ,· Part (a) follows immediately from linearity in the integrand and the results given in Examples 3 and 4. Part (b) can be done most easily from the definition. When has every Xj as a tag and at least one endpoint in each interval (Xj_ „ xj), the value of αΔ/(Φ) is precisely the expression given in (b). It suffices to define γ so that γ(ζ) contains no x} distinct from z. Then any γ-fine Φ has the properties named and the conclusion is immediate. 4. The jumps in / occur on the left-hand side of the integer points. Consequently, from Exercise 3(b), j'0a df when = Σ"-ι«(*7) » < ί < η + 1. Now /ο°α df exists if and only if lim^^/OA df exists. Consequently, the existence of the integral is equivalent to convergence of ΣΤ-Mxj)- Moreover j> 5. Let a be the primitive F of Example 2, p. 79. Since this function is continuous, it is a regulated function. It was constructed so that it is not a function of bounded variation. Since ο is a primitive of a', a2/2 is also a primitive of αα' and faaa' exists. It can be converted into J*A da. The failure of jafda to exist is the same as failure of existence of /*/<*'. Clearly, it is desirable that f(x) and 266 APPENDIX a'(x) have the same sign. The sign of a'(x) is alternately positive and negative in intervals (CQ, C,), (C„ C2),.. ·, beginning with a positive value in (c0, c,). The value of a' on (cn_„c„) is a„/(cn-cn_,) where («X., is the sequence 1, —1/2, 1/2, -1/3, 1/3,... . On [c„_,, c„) let /(x) = ft„ where (6„)f_i is the sequence 1, - 1/ln 2, 1/ln 2, - 1/ln 3, 1/ln 3,... . Then Σ"-ι«Α is divergent. Since = Σ*-ια***» ^ follows that Hm,^,bf'afa' does not exist. Consequently, /*/α' and J*/da do not exist. Finally, observe that/is regulated. It clearly has one-sided limits at each point in [a, b). Since ϋπν^οΑ = 0 the left-hand limit of / at b is also zero. 6. Let C= {c„ c2, c3,... } include all left-hand discontinuities of a, whether φ(ο„)Φ0 or not. Then ΣΒ°-ΙΦ(°ΙΙ) is absolutely convergent and there is m such that *2k-m+i\ |φ(ζ) - (α(ζ) - α(«))(/(ζ) -/(«))| < €/2» for all u such that u < ζ and u e γ(ζ). When ζ € C the function α is left-hand continuous at z. Thus there is γ(ζ) such that |(a(z) - a(«))(/(z) -/(«))| < c|/(z) -/(«)| when Μ < ζ and a G γ(ζ). When ζ $ {c,, c2,..., cm} restrict γ(ζ) further so that γ(ζ) contains none of c„ c2,..., cm. Let Φ be γ-fine. Break Φ into three subsets. Put z[u,v] into 6 when ζ £ C, into 5" when ζ e C and w < z, and into § when ζ ε C but u = z. Let be the set of all η such that c„ is a tag in iF. Then {1, 2,..., m} C £. Recall that the third restriction on γ implies that each interval of 3) which contains c„, η < m, has c„ as its tag. Whether Θ SOLUTIONS FOR CHAPTER 7 267 contains one or two intervals to which c„ belongs, one of them does not have cn as its left endpoint since α φ c„. The definition of Φ implies that Φ( <φ(5)- Σ Φ(θ| + |φ(δ)Ι ιι-J I ne.K Σ <ΚΟ ««χ For each η Ε Κ there is exactly one z[u, ν] of f with ζ = c„. Thus |Φ(£) - ΣΗΕ*<ΚΟΙ < Σ„Ε W2" < €· Recall that / is a function of bounded variation. The choice of y(z) when z$C allows us to say that |Φ(δ)| < €|Δ/|(δ) b 6 Φί^,-ΣΦίΑ) One more observation is needed to complete the proof. The set C was chosen to include all points where α is not left-hand continuous. The sum of the series Ση^ι^/ι) *s the same if some other sequence (c„)"„, is used so long as all points where φ is nonzero are included. Thus the proposition is true as stated. 7. Suppose F has bounded variation on [a, b]. According to p. 223 the variation of F satisfies VbF = lim lim \F(b) - F(t)\. Since a has bounded variation on [a, t], we already know x that Va-F = j'a\f(x)\dVa a. Moreover, hm„b\F(b) - F(t)\ b = \im,^b\j f da\. But from p. 187 we see that this last limit is zero since Δα([ί, b]) = 0. From the same source Sa\f(x)\dVa*a = ]iml^bia\f(x)\dVa*a since the variation 268 APPENDIX of a, like a, is defined on [a, ft). Now /* \f(x)\ dVxa — VbF, as required. 8. We know that ftf° τ d(a ο τ) = /;$/*/« when υ < d. For convenience set g = f ° τ and β = a ° τ. Then β is continuous at and limp_>dAj8([u, rf]) = 0. Consequently, existence of Jdgd0 is equivalent to d v existence of ton^ftgdfi and j cg dp = \imD^ cg dp. The existence of this limit can be determined by examining lim^Jl^f da. Set F(x) = /*(c)/ da. Then F is continuous at χ = r(d) since α is continuous there. The next point to note is that F ° τ is continuous at d. Thus I™/*-lim f^/A-lim (Vgdp= fW •'T(C) v-*dJr^c) v-*dJc Jc 9. Let e = ft, < c, < ft2 < c2 < · · · with limn^006„ = ft. For every η let a(ft„) = 0 and a(c„) = 2/γίΓ. Let a be linear on the intervals [ft„, c„] and [c„, bn+l\. Finally, let a(ft) = 0. On the intervals [b„, c„] where α is increasing we will assign / positive values and on [c„, ftn+1] where α is decreasing / will be negative. To be specific, let f(bn) = /(c„) = 0 for all n. At the midpoint of [b„, cn] let/ have the value l/Jn . Midway between c„ and ftn+J let/ have the value —l/Jn. Between these points let / be linear. Let /(ft) = 0. Then / is continuous on [a, ft]. Moreover /£·/ da = J%/a'. On (ft„, c„) the derivative a' is constant with value a(c„)/(cn — ft„). Moreover /£/ is the area of a triangle with altitude I/-fit and base c„ — b„. c Thus j b" f da = \ / n. A similar analysis shows that l/n. Let a„(x) = a(.t) when 0 < χ < ft„ and αη(χ) = 0 when ft„ < χ < ft. Then a„ converges uniformly to a on [a, ft]. Moreover /*/ Absolute continuity, 129 Cantor's set and function, 109, 129 Absolute integrability, 77, 91 Cartesian product, 31, 147, 151 characterized using sums, 238 Cauchy criterion for for Stieltjes integrals, 202, 225 convergence of sequences, 50 of real-valued components, 78 existence of integrals, 50 on measurable sets, 126 limits according to a direction, on unions and intersections, 94 115, 119 Chain, 216 Absolutely integrable functions, Chain rule, 60 77 Change of variables, 59 ff. approximated by step functions, in multiple integrals, 244 146 in Stieltjes integrals, 205 ff. are measurable, 136 Comparison test, 78, 115, 120, 136 Additivity, 47 for Stieltjes integrals, 203 countable, 119, 124, 128 Compatibility theorem, 16 finite, 55, 57, 114, 124 in R, 38 _ linearity implies, 68 proof in R>", 168 of Stieltjes integrals, 184 Composite function as primitive, of the total variation, 221 62 proof for integrals, 52 Composites of measurable and Almost all, 110 continuous functions, 139 Almost everywhere, 110 Continuity Arc length formula, 81 absolute, 129 Area definition, 122 in terms of open sets, 133 Area interpretation of integrals, 8, of integrals, 58 22, 122 of Stieltjes integrals, 187 Axiom of choice, 127 Continuous function as integrand in Stieltjes inte- Boas, R. P., Jr., 244, 245 grals, 194 Botts, Truman, 244, 246 as integrator, 194 Bounded convergence theorem, 90 Continuous restriction, 134 Bounded model of RA 32 Convergence Bounded variation, 79 almost everywhere, 111 270 INDEX bounded, 90 Discontinuities of dominated, 88 functions of bounded variation, monotone, 86 222 theorems for sequences, 84, 86, regulated functions, 225 88,96, 110, 112, 118,204,205 Division, 7 theorems for series, 87, 89, 112, compound, 165 ff. 119 doubly compound, 168 uniform, 83, 95 in R>, 32 Convergent series as partial, 76 an integral, 31, 36 refinement of, 77 a Stieltjes integral, 192 regular, 56 Countable additivity, 119,124, 128 subset of, 74 Countable sets tagged, 7, 17 are M-null, 108 tagged partial, 77 in the fundamental theorem, 26, Dominated convergence theorem, 109 88, 110 Countable-to-one, 63 for series, 89 Countable union of for Stieltjes integrals, 205 /M-null sets, 106 proof, 96 ff. integrable sets, 124 Dot product, 68, 227 measurable sets, 125 Φ-null sets, 159 Existence of Covering lemma, 143 integrals on subintervals, 51 Cross product, 68, 227 Stieltjes integrals on subinter- Curves vals, 183 equivalent, 215 Existence theorems for Stieltjes in- length of, 80 tegrals, 192 opposite, 215 parametric, 80, 214 Finite additivity of functions of intervals, 57 integrals, 55 Degenerate intervals are M-null, integrals on non-interval sets, 108 114 Derivative definition, 41 Stieltjes integrals, 184 Differentiability of integrals, 58, Finite points, 31 187,244 Fubini's theorem, 35, 150 Differential notation in iterated and integrable sets, 153 integrals, 151 and measurable sets, 153 Differentiation term-by-term, 85 assuming absolute integrability, Direction, 93 174 INDEX 271 on non-interval sets, 153 Generalized Riemann integral preliminaries to the proof, definition, 18 164 ff. improper extensions, 30 proof, 168 ff. Graph of continuous function is statement, 152 Af-null, 109 Function Greatest lower bound of two func- absolutely integrable, 77 ff. tions, 82 Cantor's, 109, 129 Green's theorem, 216, 220 characteristic, 123 complex-valued, 17, 185 continuous null, 128 Henstock, Ralph, 2, 245 increasing at a point, 46 Henstock's lemma, 74 measurable, 133 for Stieltjes integrals, 186 non-integrable with integrable proof, 90 absolute value, 127 Hirschman, 1.1., Jr., 245 null, 105 Holder's inequality, 143, 245 of bounded variation, 79 Hyperplane, 68 of bounded variation as inte- grand, 194 of bounded variation as integra- Inequalities tor, 194 for integrals, 49 of bounded variation is regu- Holder's, 143, 245 lated, 222 Young's, 147 piecewise monotone, 101, 207 Infimum, 82 regulated, 193 integrability of, 83 simple, 138 Infinite points, 31 step, 116 Integrability variation, 201, 221 absolute, 77, 91, 126, 202, 225 vector-valued, 17, 49, 78, 184 deduced from the fundamental Functional, 229 theorem, 28 Fundamental theorem of calculus, in McShane's sense, 237 25, 27, 109, 130 necessary conditions in R', 155 in iterated integrals, 153 of continuous functions, 57 proof, 40 ff. of lattice combinations, 83, 204 of polynomials, 28 Gauge integral, 177 of products, 141, 210 Gauge limit, 38, 177, 181 on subintervals, 51, 183 Gauges, 10 on subsets, 115 in R>, 33 sufficient conditions in Κ', on subintervals, 65 156 ff. 272 INDEX Integrable sets, 123 on expanding sequences, 118 covered by intervals, 145 on non-interval sets, 34, 113 operations on, 124 Integration by parts, 50 Integral correction term, 197 area interpretation, 8, 22, 122 for gauge integral, 199 continuity of, 58, 187 for norm integral, 196 differentiation of, 58, 187, 244 for refinement integral, 196 existence as analogue of series for Riemann integral, 195 convergence, 67 Integrators, 177 gauge, 177 continuous functions as, 194 generalized Riemann definition, functions of bounded variation 18 as, 194 improper, 28 integrals as, 186 inequalities, 49, 185 sequences of, 211 iterated, see Iterated integral Interchange of limits, 93 Lebesgue 2, 231, 234, 244 Intervals line, 214 bounded, 23 linearity in integrand, 49, 183 in R>, 31 linearity in integrator, 184 non-degenerate, 31 multiple, 31, 150 open in R', 31 norm, 177 oriented, 216 of Stieltjes type, see Stieltjes in- overlapping, 31 tegral unbounded, ?3 on an unbounded interval, 21 Iterated integral, 34, 150 on a path, 216 analogue of primitive, 158 on N, 36 as function of intervals, 155,158 oriented, 58 behavior as conditions for inte- refinement, 177 grability, 155, 159 r-fold, 152 equality of, 155 Riemann definition, 8 Iterated limits theorem, 95 two-fold, ISO zero, 49, 105 Kurzweil, Jaroslav, 2, 245 Integration component-by-component, 49 Lattice operations, 81 of absolute values, 78 on measurable functions, 140 of series of positive terms, 87 Least upper bound, 81 of series term-by-term, 37, 87, Lebesgue 112 dominated convergence theo- on closed, bounded sets, 117 rem, 88,96, 110,205 on differences of sets, 114 integral, 2, 231, 234, 244 INDEX 273 measurable sets, 231, 23S null functions, 140 measure, 231, 23S real-valued components, 134 Length sets, 125 Euclidean, 17 sets on which integrals exist, 126 of bounded intervals, 18 Measurable functions, 133 of curves, 80 algebraic operations on, 140 of unbounded intervals, 23 approximated by simple func- Limits tions, 139 according to a direction, 94 approximated by step functions, Cauchy criterion for, 94 146 left-hand, 188, 222 composed with continuous gauge, 38, 177, 181 functions, 139 interchange of, 93 sequences of, 140 iterated, 93, 95 Measurable sets, 125 norm, 177, 181 characterized, 234 of functions of bounded varia- in Lebesgue's sense, 231 tion, 222 Measure of increasing functions, 222 in Lebesgue's sense, 231, 235 of integrals over expanding in- of intervals, 33 tervals, 28, 65, 121, 130 of sets, 123 of the variation function, 223 properties, 124, 125 one-sided, 188 Monotone convergence theorem, refinement, 177, 181 86, 112 right-hand, 182, 188, 222 alternative proof, 102 uniform, 83, 95 for Stieltjes integrals, 204 Linearity of integrals, 49, 183 proof, 96 ff. Line integral, 214 Multiple integrals, 31, 150 ff. Lipschitz continuous, 214 change of variables, 244 Lp space, 245 Munroe, Μ. E., 244, 245, 246 Negative part, 82 McShane, E. J., 232, 237, 244, 245, Norm integral, 177 246 Norm limit, 177, 181 Mean value theorems for Norm of a division, 181 integrals of products, 211 Numbers Stieltjes integrals, 209 extended positive integers, 36 Measurability of extended reals, 22 Cartesian products, 147 derivatives, 141 Path, 216 functions, 133 Piecewise monotone and one-to- limits of sequences, 140 one, 101, 207 274 INDEX Point at infinity, 31 Series Positive integers augmented by oo, as an integral, 31, 36 36 as a Stieltjes integral, 192 Positive part, 82 dominated convergence theo- Primitive, 27 rem, 89 as integrand and integrator, 195 double, 171 existence, 57, 85 Fourier, 245 not of bounded variation, 79, integration term-by-term, 37, 87, 242 112 of absolutely integrable func- of positive terms, 87 tion, 79, 81 Set Product bounded, 116 Cartesian, 31, 147, 151 Cantor, 109, 129 cross, 68, 227 closed, 116 dot, 68, 227 countable set is Λί-null, 108 integrability of, 141, 210 countably infinite, 19 /M-null, 106 Refinement, 77 Ct, 234 integral, 177 integrable, 123 limit, 177, 181 inverse image of, 133 Regulated function, 193 Lebesgue measurable, 235 as integrand, 194 Lebesgue null, 233 as integrator, 194 measurable, 125 uniform limit of step functions, measurable characterized, 234 193, 224 M-null, 107, 128 Restriction of a function, 134 non-countable null, 109 Riemann integral definition, 8 non-measurable, 127 Riemann sum, 7, 23, 33 open in an interval, 116 for iterated integrals, 164 Φ-null, 159 for Stieltjes integrals, 177 Set operations in McShane's sense, 237 on integrable sets, 124 Riesz representation theorem, 230 on measurable sets, 125 Splitting intervals and sums, 53,92 Sequence of functions Step functions, 46, 143 decreasing, 86 approximate absolutely integra- increasing, 86 ble functions, 146 monotone, 86 approximate measurable func- Sequence of integrators, 211 tions, 146 Sequence of sets as integrand and integrator, 191 expanding, 118 Stieltjes integral, 177 ff. non-overlapping, 118 additivity, 184 INDEX 275 change of variables, 205 ff. Tagged intervals comparison test, 203 compatible with γ, 33 component-by-component inte- γ-fine, 33 gration, 184 in R>, 32 continuity of, 188 McShane's sense, 237 convergent series as, 192 Taylor, A. E., 244, 246 differentiability of, 188 Tonelli's theorem, 156 dominated convergence theo- Total variation, 79, 201 rem, 205 additivity of, 221 inequalities, 185 limits of, 187 Uniform convergence linearity, 183 according to a direction, 95 reduced to ordinary integral, of sequences, 83 186 Uniform convergence theorem Straddle lemma, 41 for integrals, 84 Stricter than, 12, 40 for Stieltjes integrals, 194 Subset of Uniformly continuous, 46 /M-null set, 106 Uniqueness of Φ-null set, 159 integrals, 18, 39 Substitution, 60 limits, 94 Supremum, 81 integrability of, 83 Variation function, 201, 221 Volume of 3-dimensional set, 123 Tag, 7 Weiss, Guido, 245 Tagged division, 7, 17 Williamson, J. H., 244, 245, 246 compatible with γ, 10 γ-fine, 10 Young's inequality, 147 in R> 32 McShane's sense, 237 Zero integrals, 49, 105 AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS VOL AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS VOL 20 20 The Generalized Riemann Integral Robert M. McLeod The Generalized Riemann Integral The Generalized The Generalized Riemann Integral is addressed to persons who already have an acquaintance with integrals they wish to extend and to the teachers of generations of students to come. The organization of the Riemann Integral work will make it possible for the rst group to extract the principal results without struggling through technical details which they may nd formidable or extraneous to their purposes. The technical level starts low at the opening of each chapter. Thus, readers may follow Robert M. McLeod each chapter as far as they wish and then skip to the beginning of the next. To readers who do wish to see all the details of the arguments, they are given. The generalized Riemann integral can be used to bring the full power of the integral within the reach of many who, up to now, haven't gotten a glimpse of such results as monotone and dominated conver- gence theorems. As its name hints, the generalized Riemann integral is de ned in terms of Riemann sums. The path from the de nition to theorems exhibiting the full power of the integral is direct and short. M A PRESS / MAA AMS 4-color Process 290 pages spine: 9/16" finish size: 5.5" X 8.5" 50 lb stock