Klaus Bichteler
Integration – A Functional Approach
Reprint of the 1998 Edition Klaus Bichteler Department of Mathematics The University of Texas Austin, TX 78712 USA [email protected]
1991 Mathematics Subject Classification 28-01, 28C05
ISBN 978-3-0348-0054-9 e-ISBN 978-3-0348-0055-6 DOI 10.1007/978-3-0348-0055-6
c 1998 Birkhauser¨ Verlag Originally published under the same title in the Birkhauser¨ Advanced Texts – Basler Lehrbucher¨ series by Birkhauser¨ Verlag, Switzerland, ISBN 978-3-7643-5936-2 Reprinted 2010 by Springer Basel AG
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Contents
Preface ...... ix
Chapter I Review ...... 1 I.1 Introduction ...... 1 I.2 Notation ...... 3 I.3 The Theorem of Stone–Weierstraß ...... 8 I.4 The Riemann Integral ...... 17 I.5 An Integrability Criterion ...... 21 I.6 The Permanence Properties ...... 23 I.7 Seminorms ...... 28
Chapter II Extension of the Integral ...... 31 II.1 Σ–additivity ...... 33 II.2 Elementary Integrals ...... 35 II.3 The Daniell Mean ...... 42 II.4 Negligible Functions and Sets ...... 48 II.5 Integrable Functions ...... 52 II.6 Extending the Integral ...... 58 II.7 Integrable Sets ...... 61 II.8 Example of a Non–Integrable Function ...... 66
Chapter III Measurability ...... 69 III.1 Littlewood’s Principles ...... 70 III.2 The Permanence Properties ...... 73 III.3 The Integrability Criterion ...... 75 III.4 Measurable Sets ...... 81 III.5 Baire and Borel Functions ...... 85 III.6 Further Properties of Daniell’s Mean ...... 92 III.7 The Procedures of Lebesgue and Carath´edory ...... 95 viii Contents
Chapter IV The Classical Banach Spaces ...... 101 IV.1 The p–norms ...... 102 IV.2 The Lp –spaces ...... 105 IV.3 The Lp–spaces ...... 108 IV.4 Linear Functionals ...... 112 IV.5 The Dual of Lp ...... 118 IV.6 The Hilbert space L2 ...... 122
Chapter V Operations on Measures ...... 125 V.1 Products of Elementary Integrals ...... 125 V.2 The Theorems of Fubini and Tonelli ...... 130 V.3 An Application: Convolution ...... 133 V.4 An Application: Marcinkiewicz Interpolation ...... 134 V.5 Signed Measures ...... 137 V.6 The Space of Measures ...... 144 V.7 Measures with Densitites ...... 151 V.8 The Radon–Nikodym Theorem ...... 154 V.9 An Application: Conditional Expectation ...... 155 V.10 Differentiation ...... 157
Appendix A Answers to Selected Problems ...... 165
References ...... 180
Index of Notations ...... 181
Index ...... 184 Preface
This text originated as notes for the introductory graduate Real Analysis course at the University of Texas at Austin. They were intended for students who have had a first course in Real Analysis ( −δ–proofs) and know the topology of the real line, in particular the notion of compactness, and the Riemann integral. The main topic in such a course traditionally is integration theory, and so it is here. The approach taken is different from the usual one, though. Usually the development follows history. First the content is extended to a measure on a σ–algebra, — which is identified through Carath´eodory’s cut condition — then to the integrable functions through another set of limit operations. Daniell observed that the extensions have much in common and can be done in one effort, saving half the labor. We follow in his footsteps. The emphasis is not, however, on saving time so much as it is on functionality: we ask what the purpose and expected properties of the prospective integral are and then fashion definitions and procedures accordingly—history and the usual treatment are reviewed, though, in Section III.7, so as to give the reader the connection of the historical development with the present one. Functionality also guides the treatment of measurability. The text does not set down the notion of a σ–algebra as a gift from our betters, followed by a (mysterious) definition of measurability on them; this would pressure the gentle reader into accepting authoritarian mathematics or leave the rebellious one with many questions: where do these definitions come from, where do they lead, and how did anyone ever dream them up? Rather, we ask what the obstruction to the integrability of a function might be besides being too big, in other words, what the local structure of the integrable functions is. Two simple observations in answer to this lead to Littlewood’s Principles, which are used to define measurability. The permanence properties of measurability are rather easy to establish this way, and once they are available so is the fact that the definition agrees with the traditional one. A third uncommon aspect of the development is the use of seminorms. Instead of arguing with the upper and lower integrals of Daniell, analogs of Lebesgue’s outer and inner measure, we observe that already in the case of the Riemann integral a simple absolute–value sign under the upper integral simplifies matters considerably, by rendering superfluous the study of the lower integral and pro- viding a seminorm, the Daniell mean. The analyst will often attack a problem by defining a suitable seminorm with which to identify a function space in which a solution reasonably can be sought. It seems to me that it is not too early to introduce this tool in the context of integration: the problem is to extend the ele- mentary integral from step functions to larger classes, and the Jordan and Daniell means are perfectly suitable seminorms towards that goal. x Preface
The exposition starts with a review of Weierstraß’ theorem and the Riemann integral. The latter review is designed to show that the integral of Lebesgue– Carath´eodory emerges from rather a minute alteration to Riemann’s; it is not the mystery but the plethora of wonderful properties of the new integral that ask for another 120 pages just to write them down and prove them. The exercises are an integral part of the material. Those marked by an asterisk ∗ are used later on and thus must be done. For the majority of them there are answers or at least hints in the appendix. The first version of these notes was replete with mistakes, and I thank my students for pointing them out. I hope the remaining mistakes are small in number; they are all mine.
Klaus Bichteler The University of Texas at Austin //www.ma.utexas.edu/users/kbi [email protected] May 1998 Chapter I Review
I.1 Introduction
Riemann’s integral is a major achievement of civilization. It solves in a rigourous way the ancient problems of squaring the disk and straightening the circle. It provides a solid foundation for Newtonian Physics, on which modern theories of physics and engineering can grow. Nevertheless, it has shortcomings: its limit theorems are not good enough, and not sufficiently many functions are Riemann– integrable. Consequently it is an insufficient tool for quantum physics and the associated partial differential equations. These shortcomings can be overcome, though, by a careful analysis of its concepts and a surprisingly slight improvement on them. The integral of a step function whose steps are intervals is the sum of the products step–size×height–of–step. Riemann’s well–known procedure to integrate a function f that is not a step function is to squeeze it with step–functions from above and below: One defines the upper Riemann integral of f to be the infimum of the integrals of step–functions above f and its lower Riemann integral to be the supremum of the integrals of step–functions below. Then one says that f is Riemann integrable if these two numbers agree, and in that case defines the Riemann integral of f as the common value. We shall refer to this idea as the “Riemann squeeze.” Lebesgue’s original idea towards enlarging the class of integrable functions was this: if at the outset more sets than just intervals could be measured then the class of step functions, to which an integral could be assigned by the formula sum of the products step–size×height–of–step, would be richer; more functions would be available to do the squeezing with; thus the Riemann–squeeze could be applied to more functions. This would result in a larger class of integrable functions. Lebesgue did, indeed, succeed in measuring many more sets. It turns out, however, that the Riemann squeeze by itself is not adequate even then, and that further limit operations have to be performed in order to turn this idea into a complete success — the main reason being that the Riemann construction of the upper and lower integrals of f still requires the absolute value |f| of the integrand f to be majorized by a step function, i.e., to be bounded and to vanish outside some integrable set.
K. Bichteler, Integration – A Functional Approach, Modern Birkhäuser Classics, DOI 10.1007/978-3-0348-0055-6_1, © Birkhäuser Verlag 1998 2Review
Daniell noticed that Lebesgue’s original extension of the measure from in- tervals to his larger class of integrable sets and the subsequent limit operations had much in common and that half the labour can be saved by combining the two efforts. Daniell’s method consists in replacing the Riemann upper and lower inte- ∗ grals by slightly different ones, termed the Daniell upper and lower integrals and ∗ , and then to squeeze as before. We shall follow his path to Lebesgue’s integral, modified minutely by the following observation: A Riemann integrable function f is evidently close to being a step function itself, since it can be squeezed arbitrarily closely by the latter. This somewhat vague statement can be given a precise meaning by the following notion of the natural distance of f from a step function φ: dist (f,φ)= f − φ = |f − φ| .
A simple application of the triangle inequality shows that f is Riemann integrable if and only if there is a sequence (φn) of step functions whose distance f − φn from f goes to zero. The integral of f is then the limit of the integrals of the φn . The size measurement is a seminorm. Similarly, Lebesgue integrable are pre- cisely those functions f whose distance from step functions φ as measured by the Daniell mean ∗ , ∗ ∗ dist (f,φ)= f − φ ∗ = |f − φ| , can be made as small as one pleases. Again the integral is the extension by continuity. Looking at the integral this way, namely, as an extension by con- tinuity under a suitable seminorm, reduces the technicalities some more and is in keeping with modern analysis, where seminorms and the notion of ex- tension by continuity play an all–pervading rˆole. Furthermore, it aims right at the heart of the problem: measure theory is, despite its name, not so much about measuring sets as about sizing functions. (This is not meant to dis- parage the measurement of sets; most frequently, a nuts–and–bolts problem about estimating the mean, or another gauge of the size, of a function f in- volves estimating the measure of the set of points where f is large or small or has some other property of interest — see for instance Sections V.4 and V.10.) After some notation has been established, this chapter continues with a section on the theorem of Weierstraß. A short review of the Riemann integral follows. It is explained in detail how the Riemann integral can be viewed as the extension of a linear map under a suitable seminorm. We take the occasion to explain this important notion guided by that familiar example. Section I.2 Notation 3
I.2 Notation
The Reals. The reader is of course familiar with the field R of real numbers. It is the one and only complete ordered field. It contains the natural numbers N = {0, 1, 2,...}, the ring of integers Z = {...,−2, −1, 0, 1, 2,...},andthe field Q of rational numbers. The rationals are dense in the reals. R is also called the real line. The reader not familiar with any of these notions should consult her textbook on undergraduate Real Analysis or [9, pp. 31–50]. Occasionally the complex number field C comes into the picture.
Positivity and Negativity. Arealnumbera will be called positive if a ≥ 0, and R+ denotes the set of positive reals. Similarly, a real–valued function f is positive if f(x) ≥ 0 for all points x in its domain dom(f). If we want to stress that f be strictly positive: f(x) > 0 ∀ x ∈ dom(f), we shall say so. It is clear what the words “negative”and“strictly negative”mean.IfF is a collection of functions then F+ will denote the positive functions in F ,etc.Notethata positive function may be zero on a large set, in fact, everywhere. The statements “b exceeds a,” “b is bigger than a,” and “a is less than b”allmean“a ≤ b;” modified by the word “strictly” they mean “a Exercise 2.1 To practice this convention let f,g be two functions having the same domain. Write the negation of the statement “g exceeds f .” The Larger or Smaller of Two Real Numbers a, b is customarily denoted by a ∨ b or a ∧ b, respectively. Similarly, the pointwise maximum or minimum of two functions f,g is denoted by f ∨ g and f ∧ g , respectively: f ∨ g (x)=max{f(x),g(x)} or f ∧ g (x)=min{f(x),g(x)} . | | →| | ∨ − ∨ − f is the function x f(x) ;andf+ = f 0, f− =( f) 0, so that f = f+ f− . F F Suppose is a family of functions. Then = f∈F f is their pointwise supremum: F (x)=supf(x);and F (x)= inf f(x) f∈F f∈F is their pointwise infimum. Very frequently, the family F is countable: F = {f1,f2,...}.Thenwewrite fn (x)=supfn(x)and fn (x)=inffn(x) , n∈N n n∈N n 4Review respectively. Recall that the limit inferior and the limit superior of a sequence (an) of reals are defined as lim inf an = lim inf an = an and n→∞ N→∞ n≥N ∈N ≥ N n N lim sup an = lim sup an = an , respectively. n→∞ N→∞ n≥N N∈N n≥N Exercise 2.2 For all a, b, c, r ∈ R,(a∧b)+c =(a+c)∧(b+c)and(a∨b)+c =(a+c)∨(b+c); ∨ ∨ ≥ ∨ ∧ ≤ ∧ ∨ 1 | | r(a b)=ra rb if r 0andr(a b)=ra rb if r 0; a b+a b = a+b, a+ = 2 ( a +a), 1 1 1 − | |− ∨ | − | ∧ −| − | and a = 2 ( a a); a b = 2 (a + b + a b )anda b = 2 (a + b a b ). The Extended Real Line R isthereallineaugmentedbythetwosymbols−∞ and ∞ =+∞: R = {−∞} ∪ R ∪{+∞} . We adopt the usual conventions concerning the arithmetic and order structure of the extended reals R: −∞