Real Analysis

1 Real AnalysisCourse NotesC. McMullenContents1 Introduction ..12 Set Theory and the Real Numbers ..43 Lebesgue Measurable Sets .. 134 Measurable Functions .. 265 Integration .. 356 Differentiation and Integration .. 447 The Classical Banach Spaces .. 608 Baire Category .. 729 General Topology .. 8110 Banach Spaces .. 9711 Fourier Series .. 11212 Harmonic Analysis onRandS2.. 12613 General Measure Theory .. 131 AMeasurableAwithA Anonmeasurable .. 1361 IntroductionWe begin by discussing the motivation for real Analysis , and especially forthe reconsideration of the notion of integral and the invention of Lebesgueintegration, which goes beyond the Riemannian integral familiar from clas-sical of one of the oldest branches of mathematics,and one that includes calculus, Analysis is hardly in need of just in case, we remark that its uses include:1.

2 The description of physical systems, such as planetary motion, bydynamical systems (ordinary differential equations);2. The theory of partial differential equations, such as those describingheat flow or quantum particles;3. Harmonic Analysis on Lie groups, of whichRis a simple example;4. Representation theory;15. The description of optimal structures, from minimal surfaces to eco-nomic equilibria;6. The foundations of probability theory;7. Automorphic forms and analytic number theory; and8. Dynamics and ergodic now motivate the need for a sophisticated theoryof measure and integration, called the Lebesgue theory, which will form thefirst topic in this Analysis it is necessary to take limits; thus one is naturally led tothe construction of the real numbers, a system of numbers containing therationals and closed under limits.

3 When one considers functions it is againnatural to work with spaces that are closed under suitable limits. For exam-ple, consider the space of continuous functionsC[0,1]. We might measurethe size of a function here by f 1= 10|f(x)|dx.(There is no problem defining the integral, say using Riemann sums).But we quickly see that there are Cauchy sequences of continuous func-tions whose limit, in this norm, are discontinuous. So we should extendC[0,1] to a space that is closed under limits. It is not at first even evidentthat the limiting objects should befunctions.

4 And if we try to includeallfunctions, we are faced with the difficult problem of integrating a modern solution to this natural issue is to introduce the idea ofmeasurable functions, a space of functions that is closed under limits andtame enough to integrate. The Riemann integral turns out to be inadequatefor these purposes, so a new notion of integration must be invented. In factwe must first examine carefully the idea of the mass ormeasureof a subsetA R, which can be though of as the integral of its indicator function A(x) = 1 ifx Aand = 0 ifx6 classical motivation for the Lebesgue integralcome from Fourier : [0, ] Ris a reasonable function.

5 We define the Fouriercoefficients offbyan=2 0f(x) sin(nx) the factor of 2/ is chosen so that2 0sin(nx) sin(mx)dx= observe that iff(x) = 1bnsin(nx),then at least formallyan=bn(this is true, for example, for a finite sum).This representation off(x) as a superposition of sines is very useful forapplications. For example,f(x) can be thought of as a sound wave, whereanmeasures the strength of the what coefficientsancan occur? The orthogonality relation impliesthat2 0|f(x)|2dx= |an| makes it natural to ask if, conversely, for anyansuch that |an|2< ,there exists a functionfwith these Fourier coefficients.

6 The natural functionto try isf(x) = ansin(nx).But why should this sum even exist? The functions sin(nx) are onlybounded by one, and |an|2< is much weaker than |an|< .One of the original motivations for the theory of Lebesgue measure andintegration was to refine the notion of function so that this sum reallydoes exist. The resulting functionf(x) however need to be Riemann inte-grable! To get a reasonable theory that includes such Fourier series, Cantor,Dedekind, Fourier, Lebesgue, etc. were led inexorably to a re-examinationof the foundations of real Analysis and of mathematics itself.

7 The theorythat emerged will be the subject of this are a few additional points about this , we could try to define the required space of functions calledL2[0, ] to simply be the metric completion of, sayC[0, ] with respecttod(f,g) = |f g|2. The reals are defined from the rationals in a similarfashion. But the question would still remain, can the limiting objects bethought of as functions?Second, the set of pointE Rwhere ansin(nx) actually converges isliable to be a very complicated set not closed or open, or even a countableunion or intersection of sets of this form.

8 Thus to even begin, we must havea good understanding of subsets , even if the limiting functionf(x) exists, it will generally not beRiemann integrable. Thus we must broaden our theory of integration to3deal with such functions. It turns out this is related to the second point we must again find a good notion for the length ormeasurem(E) of a fairlygeneral subsetE R, sincem(E) = Set Theory and the Real NumbersThe foundations of real Analysis are given by set theory, and the notion ofcardinality in set theory, as well as the axiom of choice, occur frequently inanalysis.

9 Thus we begin with a rapid review of this theory. For more detailssee, [Hal]. We then discuss the real numbers from both the axiomaticand constructive point of view. Finally we discuss open sets and Borel some sense, real Analysis is a pearl formed around the grain of sandprovided by paradoxical sets. These paradoxical sets include sets that haveno reasonable measure, which we will construct using the axiom of axioms of set is a brief account of the axioms. Axiom I. (Extension) A set is determined by its elements. That is, ifx A= x Band vice-versa, thenA=B.

10 Axiom II. (Specification) IfAis a set then{x A:P(x)}is also aset. Axiom III. (Pairs) IfAandBare sets then so is{A,B}. From thisaxiom and = 0, we can now form{0,0}={0}, which we call 1; andwe can form{0,1}, which we call 2; but we cannot yet form{0,1,2}. Axiom IV. (Unions) IfAis a set, then A={x: B,B A&x B}is also a set. From this axiom and that of pairs we can form {A,B}=A B. Thus we can definex+=x+ 1 =x {x}, andform, for example, 7 ={0,1,2,3,4,5,6}. Axiom V. (Powers) IfAis a set, thenP(A) ={B:B A}is also aset. Axiom VI.