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Measure and Integral Measure and integral E. Kowalski ETH Zurich¨ [email protected] Contents Preamble 1 Introduction 2 Notation 4 Chapter 1. Measure theory 7 1.1. Algebras, σ-algebras, etc 8 1.2. Measure on a σ-algebra 14 1.3. The Lebesgue measure 20 1.4. Borel measures and regularity properties 22 Chapter 2. Integration with respect to a measure 24 2.1. Integrating step functions 24 2.2. Integration of non-negative functions 26 2.3. Integrable functions 33 2.4. Integrating with respect to the Lebesgue measure 41 Chapter 3. First applications of the integral 46 3.1. Functions defined by an integral 46 3.2. An example: the Fourier transform 49 3.3. Lp-spaces 52 3.4. Probabilistic examples: the Borel-Cantelli lemma and the law of large numbers 65 Chapter 4. Measure and integration on product spaces 75 4.1. Product measures 75 4.2. Application to random variables 82 4.3. The Fubini{Tonelli theorems 86 4.4. The Lebesgue integral on Rd 90 Chapter 5. Integration and continuous functions 98 5.1. Introduction 98 5.2. The Riesz representation theorem 100 5.3. Proof of the Riesz representation theorem 103 5.4. Approximation theorems 113 5.5. Simple applications 116 5.6. Application of uniqueness properties of Borel measures 119 5.7. Probabilistic applications of Riesz's Theorem 125 Chapter 6. The convolution product 132 6.1. Definition 132 6.2. Existence of the convolution product 133 6.3. Regularization properties of the convolution operation 137 ii 6.4. Approximation by convolution 140 Bibliography 146 iii Preamble This course is an English translation and adaptation with minor changes of the French lecture notes I had written for a course of integration, Fourier analysis and probability in Bordeaux, which I taught in 2001/02 and the following two years. The only difference worth mentioning between these notes and other similar treat- ments of integration theory is the incorporation \within the flow" of notions of probability theory (instead of having a specific chapter on probability). These probabilistic asides { usually identified with a grey bar on the left margin { can be disregarded by readers who are interested only in measure theory and integration for classical analysis. The index will (when actually present) highlight those sections in a specific way so that they are easy to read independently. Acknowledgements. Thanks to the students who pointed out typos and inacurra- cies in the text as I was preparing it, in particular S. Tornier. 1 Introduction The integral of a function was defined first for fairly regular functions defined on a closed interval; its two fundamental properties are that the operation of indefinite integration (with variable end point) is the operation inverse of differentiation, while the definite integral over a fixed interval has a geometric interpretation in terms of area (if the function is non-negative, it is the area under the graph of the function and above the axis of the variable of integration). However, the integral, as it is first learnt, and as it was defined until the early 20th century (the Riemann integral), has a number of flaws which become quite obvious and problematic when it comes to dealing with situations beyond those of a continuous func- tion f :[a; b] ! C defined on a closed interval in R. To describe this, we recall briefly the essence of the definition. It is based on ap- proaching the desired integral Z b f(t)dt a with Riemann sums of the type N X S(f) = (yi − yi−1) sup f(x) y x y i=1 i−16 6 i where we have a subdivision (0.1) a = y0 < y1 < ··· < yN = b of [a; b]. Indeed, if f is Riemann-integrable, such sums will converge to the integral when a sequence of subdivisions where the steps max jyi − yi−1j converge to 0. Here are some of the difficulties that arise in working with this integral. • Compatibility with other limiting processes: let (fn) be a sequence of (continu- ous) functions on [a; b] \converging" to a function f, in some sense. It is not always true that Z b Z b (0.2) lim fn(t)dt = f(t)dt; n!+1 a a and in fact, with Riemann's definition, f may fail to be integrable. The problem, as often in analysis, is to justify a certain exchange of two limits, and this turns out to be quite difficult. The only case where the formula can be shown to hold very easily is when fn ! f uniformly on [a; b]: then f is continuous, and (0.2) holds. However, it is often very hard to check uniform convergence, and it often fails to hold. This becomes even more of a problem when working with integrals over unbounded sets like ]−1; +1[, which are themselves limits of integrals over ]a; b[ as a ! −∞ and b + 1: in effect, we then have three limiting processes to juggle. 2 The following simple example illustrates that one will always need some care: let [a; b] = [0; 1] and 8 2n2x if 0 x 1 <> 6 6 2n 2 1 1 (0.3) fn(x) = 2n − 2n x if 2n 6 x 6 n :> 1 0 if n 6 x 6 1 so that fn(x) ! 0 for all x 2 [0; 1] (since fn(0) = 0 and the sequence becomes constant for all n > x−1 if x 2]0; 1]), while Z 1 fn(x)dx = 1 0 for all n. • Multiple integrals: just like that integral Z b f(t)dt a has a geometric interpretation when f(t) > 0 on [a; b] as the area of the plane domain situated between the x-axis and the graph of f, one is tempted to write down integrals with more than one variable to compute the volume of a solid in R3, or the surface area bounded by such a solid (e.g., the surface of a sphere). Riemann's definition encounters very serious difficulties here because the lack of a natural ordering of the plane makes it difficult to find suitable analogues of the subdivisions used for one-variable integration. And even if some definition is found, the justification of the expected formula of exchange of the variables of integration Z b Z d Z d Z b (0.4) f(x; y)dxdy = f(x; y)dydx a c c a is usually very difficult. • Probability: suppose one wants to give a precise mathematical sense to such intuitive statements as \a number taken at random between 0 and 1 has proba- bility 1=2 of being > 1=2"; it quickly seems natural to say that the probability that x 2 [0; 1] satisfy a certain property P is Z b (b − a) = dx a if the set I(P) of those x 2 [0; 1] where P holds is equal to the interval [a; b]. Very quickly, however, one is led to construct properties which are quite natural, yet I(P) is not an interval, or even a finite disjoint union of intervals! For example, let P be the property: \there is no 1 in the digital expansion x = 0; d1d2 ··· of x in basis 3, di 2 f0; 1; 2g." What is the probability of this? • Infinite integrals: as already mentioned, in Riemann's method, an integral like Z +1 f(t)dt a is defined a posteriori as limit of integrals from a to b, with b ! +1. This implies that all desired properties of these integrals be checked separately { and involves therefore an additional limiting process. This becomes even more problematic 3 when one tries to define and study functions defined by such integrals, like the Fourier transform of a function f : R ! C by the formula Z f^(t) = f(x)e−ixtdx −∞ for t 2 R. H. Lebesgue, who introduced the new integration that we will study in this course as the beginning of the 20th century used to present the fundamental idea of his method as follows. Suppose you wish to compute the sum S (seen as a discrete analogue of the integral) of finitely many real numbers, say a1; a2; : : : ; an (which might be, as in the nice illustration that Lebesgue used, the successive coins found in your pocket, as you try to ascertain your wealth). The \Riemann" way is to take each coin in turn and add their values one by one: S = a1 + ··· + an: However, there is a different way to do this, which in fact is almost obvious (in the example of counting money): one can take all the coins, sort them into stacks correspond- ing to the possibles values they represent, and compute the sum by adding the different values multiplied by the number of each of the coins that occur: if b1,..., bm are the different values among the ai's, let Nj = number of i 6 n with ai = bj; denote the corresponding multiplicity, and then we have S = b1 × N1 + ··· + bm × Nm: One essential difference is that in the second procedure, the order in which the coins appear is irrelevant; this is what will allow the definition of integration with respect to a measure to be uniformly valid and suitable for integrating over almost arbitrary sets, n and in particular over any space R with n > 1. To implement the idea behind Lebesgue's story, the first step, however, is to be able to give a meaning to the analogue of the multiplicities Nj when dealing with functions taking possibly infinitely many values, and defined on continuous sets.
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