DOCSLIB.ORG

MEASURE THEORY Course

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
MEASURE THEORY Course MEASURE THEORY ARIEL YADIN Course: 201.1.0081 Fall 2014-15 Lecture notes updated: January 22, 2015 (partial solutions) Contents Lecture 1. Introduction 4 1.1. Measuring things 4 1.2. Elementary measure 5 This lecture has 6 exercises. 11 Lecture 2. Jordan measure 12 2.1. Jordan measure 12 This lecture has 15 exercises. 24 Lecture 3. Lebesgue outer measure 25 3.1. From finite to countable 25 This lecture has 5 exercises. 29 Lecture 4. Lebesgue measure 30 4.1. Definition of Lebesgue measure 30 4.2. Lebesgue measure as a measure 37 This lecture has 12 exercises. 43 Lecture 5. Abstract measures 44 5.1. σ-algebras 44 5.2. Measures 47 5.3. Fatou's Lemma and continuity 50 This lecture has 13 exercises. 52 Lecture 6. Outer measures 53 1 2 6.1. Outer measures 53 6.2. Measurability 54 6.3. Pre-measures 57 This lecture has 12 exercises. 62 Lecture 7. Lebesgue-Stieltjes Theory 63 7.1. Lebesgue-Stieltjes measure 63 7.2. Regularity 68 7.3. Non-measureable sets 69 7.4. Cantor set 69 This lecture has 6 exercises. 70 Lecture 8. Functions of measure spaces 71 8.1. Products 71 8.2. Measurable functions 73 8.3. Simple functions 78 This lecture has 20 exercises. 80 Lecture 9. Integration: positive functions 81 9.1. Integration of simple functions 81 9.2. Integration of positive functions 83 This lecture has 7 exercises. 89 Lecture 10. Integration: general functions 90 10.1. Real valued functions 90 10.2. Complex valued functions 91 10.3. Convergence 93 10.4. Riemann vs. Lebesgue integration 97 This lecture has 10 exercises. 101 Lecture 11. Product measures 102 11.1. Sections 105 11.2. Product integrals 107 3 11.3. The Fubini-Tonelli Theorem 114 This lecture has 19 exercises. 120 Lecture 12. Change of variables 121 12.1. Linear transformations of Lebesgue measure 121 12.2. Change of variables formula 124 This lecture has 4 exercises. 128 Lecture 13. Lebesgue-Radon-Nykodim 129 13.1. Signed measures 129 13.2. The Lebesgue-Radon-Nikodym Theorem 136 This lecture has 27 exercises. 146 Lecture 14. Convergence 147 14.1. Modes of convergence 147 14.2. Uniform and almost uniform convergence 153 This lecture has 10 exercises. 157 Lecture 15. Differentiation 158 15.1. Hardy-Littlewood Maximal Theorem 158 15.2. The Lebesgue Differentiation Theorem 161 15.3. Differentiation and Radon-Nykodim derivative 162 This lecture has 6 exercises. 164 Lecture 16. The Riesz Representation Theorem 165 16.1. Compactly supported functions 165 16.2. Linear functionals 168 16.3. The Riesz Representation Theorem 169 This lecture has 5 exercises. 174 Total number of exercises: 177 174 4 Measure Theory Ariel Yadin Lecture 1: Introduction 1.1. Measuring things Already the ancient Greeks developed a theory of how to measure length, area, and volume and area of 1; 2 and 3 dimensional objects. In this setting (i.e. in Rd for d 3) ≤ it stands to reason that the \size" or \measure" of an object must satisfy some basic axioms: If m(A) is the measure of a set A, it should be the same for any reflection, • translation or rotation of A. That is, m(A) = m(A + x) = m(UA) where U is a rotation matrix. For example, [0; 1]2 should heave the same measure as [ 2; 1]2 which should n o − − have the same measure as (x; y): x + y 1 , the diamond of side-length j j j j ≤ p2 1. If A can be broken into disjoint pieces, the sum of their measures should be the • measure of A. That is, m(A B) = m(A) + m(B). ] Example: [0; 1] [0; 2] should have measure that is the sum of the measures × of [0; 1]2 and [0; 1] (1; 2]. × We use to denote disjoint union; that is, A B is not only notation for a set, X ] ] but this notation claims that A B = . The small + sign remind us of the additive \ ; property above. This is already quite fruitful. If the unit square (0; 1]2 is of measure 1, then: We can determine the measure of any square of rational side length. • n X m((0; n]2) = m((j 1; j] (k 1; k]) = n2 − × − j;k=1 5 and n 2 X j 1 j k 1 k 2 1 2 m((0; 1] ) = m(( − ; ] ( − ; ]) = n m((0; ] ): n n × n n n j;k=1 We can measure any rectangle of rational side length: decompose it into squares. • We can measure right-angle triangles: disjoint union of two is a rectangle. • We can then measure any triangle, by bounding it in a rectangle and subtracting • the excess right-angle triangles. With triangles we can measure any polygon. • X What about measuring a disc? 1.2. Elementary measure Let us first formally define the above. Definition 1.1 (Boxes). Consider Rd for some d 1. • ≥ I R is an interval if I is one of [a; b]; (a; b); (a; b]; [a; b) for some < a b < • ⊂ −∞ ≤ . Note that we allow the singleton a = [a; a] as an interval, and the empty set 1 f g = (a; a). The length or measure of such I is defined to be I = m(I) = b a. ; j j − d B R is a box if B = I1 Id where Ij are intervals. The volume or • ⊂ × · · · × measure of such a box B is defined to be B = m(B) = I I . j j j 1j · · · j dj d E R is an elementary set if E = B1 Bn for some finite number of • ⊂ [···[ boxes. d d 0 = 0(R ) denotes the set of elementary sets in R . •E E Exercise 1.1. Show that is closed under finite unions, finite intersections, E0 set-difference, symmetric difference and translations. That is, show that if E; F are elementary sets then so are: E F and E F , • [ \ E F := x E : x F = E F c, • n f 2 62 g \ E F := (E F ) (F E), • 4 n [ n 6 E + x := y + x : y E . • f 2 g Exercise 1.2. Show that if E is an elementary set then there exist B1;:::;Bn pairwise disjoint boxes such that E = B B . 1 ]···] n It would be tempting to define the measure of E = B B as m(B )+ +m(B ). 1]· · ·] n 1 ··· n But we are not guarantied that the decomposition is a unique one. Proposition 1.2 (Discretisation Formula). Let I be an interval. Then, • 1 m(I) = lim #(I 1 ): n n Z !1 n \ Consequently, if B Rd is a box then, ⊂ 1 m(B) = lim #(B 1 d); n d n Z !1 n \ d and if E = B1 Bn R is an elementary set then ]···] ⊂ 1 1 d m(B1) + + m(Bn) = lim #(E ): n d n Z ··· !1 n \ Proof. Let a < b and I be an interval with endpoints a; b. For n large enough, note that 1 z #(I Z) = # z Z : I # z Z : na z nb \ n 2 n 2 ≤ f 2 ≤ ≤ g # z Z : na z nb nb + 1 (na 1) + 1 = n(b a) + 3: ≤ f 2 b c ≤ ≤ d eg ≤ − − − Similarly, #(I 1 Z) # z Z : na < z < nb n(b a) 1: \ n ≥ f 2 g ≥ − − Dividing by n and taking n we obtain the formula for intervals. ! 1 Now, if B = I I then 1 × · · · × d 1 d 1 1 B Z = (I1 Z) (Id Z); \ n \ n × · · · × \ n 7 which implies that d 1 d Y 1 #(B Z ) = #(Ij Z): \ n \ n j=1 Dividing by nd and taking n gives the formula for boxes. ! 1 The final assertion is a consequence of the fact that if A B = then \ ; d d d (A B) 1 Z = (A 1 Z ) (B 1 Z ): ] \ n \ n ] \ n ut Definition 1.3. If E = B B is an elementary set we define the measure of • 1 ]···] n E to be 1 1 m(E) = m(B1) + + m(Bn) = lim #(E ); n d n Z ··· !1 n \ which is well defined by Proposition ??. Exercise 1.3. Give an example of a set A [0; 1] such that the limit ⊂ 1 lim #(A 1 ) n n Z !1 n \ does not exist. 1 1 lim #(A 1 ) and lim #((A + x) 1 ) n n Z n n Z !1 n \ !1 n \ both exist but are not equal. Solution to ex:1.3. :( | For the first example: Set 2m 1 n A = n− : Z n; m 1 ; 2m 1 2 : 2 3 ≥ − ≤ Since for any z 1 we can write z = (2m 1)2k for some k 0 and m 1, we have ≥ − ≥ ≥ that n z n n #(A 2− Z) # n : 1 z 2 = 2 : \ ≥ 2 ≤ ≤ 8 So 1 1 n n lim sup n #(A n Z) lim sup 2− #(A 2− Z) = 1: n \ ≥ n \ !1 !1 k 2m 1 z On the other hand, if x A 3− Z then for some m; n 1 and z Z, x = n− = k 2 \ ≥ 2 2 3 which implies that 2nz = 3k(2m 1). The left hand side is even while the right hand − k side is odd, which is a contradiction. So A 3− Z = for all k, and we conclude that \ ; 1 1 n n lim inf #(A ) lim inf 3− #(A 3− ) = 0: n n n Z n Z !1 \ ≤ !1 \ For the second example take A = Q [0; 1] and x = π.:) \ X This last exercise shows that we do not want to use the discretization formula to define the measure of general sets, but we can use it for more than just elementary sets.
Load more

Recommended publications
  • An Introduction to Measure Theory Terence
    An introduction to measure theory Terence Tao Department of Mathematics, UCLA, Los Angeles, CA 90095 E-mail address: [email protected] To Garth Gaudry, who set me on the road; To my family, for their constant support; And to the readers of my blog, for their feedback and contributions. Contents Preface ix Notation x Acknowledgments xvi Chapter 1. Measure theory 1 x1.1. Prologue: The problem of measure 2 x1.2. Lebesgue measure 17 x1.3. The Lebesgue integral 46 x1.4. Abstract measure spaces 79 x1.5. Modes of convergence 114 x1.6. Differentiation theorems 131 x1.7. Outer measures, pre-measures, and product measures 179 Chapter 2. Related articles 209 x2.1. Problem solving strategies 210 x2.2. The Radamacher differentiation theorem 226 x2.3. Probability spaces 232 x2.4. Infinite product spaces and the Kolmogorov extension theorem 235 Bibliography 243 vii viii Contents Index 245 Preface In the fall of 2010, I taught an introductory one-quarter course on graduate real analysis, focusing in particular on the basics of mea- sure and integration theory, both in Euclidean spaces and in abstract measure spaces. This text is based on my lecture notes of that course, which are also available online on my blog terrytao.wordpress.com, together with some supplementary material, such as a section on prob- lem solving strategies in real analysis (Section 2.1) which evolved from discussions with my students. This text is intended to form a prequel to my graduate text [Ta2010] (henceforth referred to as An epsilon of room, Vol. I ), which is an introduction to the analysis of Hilbert and Banach spaces (such as Lp and Sobolev spaces), point-set topology, and related top- ics such as Fourier analysis and the theory of distributions; together, they serve as a text for a complete first-year graduate course in real analysis.
    [Show full text]
  • Jordan Measurability
    Jordan Measurability November 16, 2006 A bounded set E in the plane is Jordan Measurable if χE is Riemann integrable. χE is discontinuous exactly on ∂E, so from a general theorem, we have Theorem 1. A bounded set E is Jordan measurable if and only if the Lebesgue measure of ∂E is 0. However there is a better theorem: Theorem 2. A bounded set E is Jordan measurable if and only if the Jordan measure of ∂E is 0. Corollary 1. The boundary of a bounded set is of Lebesgue measure 0 if and only if it is of Jordan measure 0. The corollary can be proved directly using the Heine-Borel theorem. To prove Theorem 2 we start with a lemma. Lemma 1. A set E is of Jordan measure 0 if and only if for every > 0 there is a finite union of rectangles, n n n [ [ X Ri, with sides parallel the the axis lines, so that E ⊂ Ri and |Ri| < . 1 1 1 Proof. If E has Jordan measure 0 then the upper sums SP (χE) can be made as small as we please. This gives a finite set of rectangles satisfying the requirement. On the other had if we have a set of rectangles n n X [ with |Ri| < /2 and E ⊂ Ri, then by fattening them up slightly we can assume they are open. Then 1 1 taking a partition P that makes all edges of these rectangles unions of rectangles in the partition, we find that we can make SP (χE) < . Proof. (of Theorem 2.) Suppose E is Jordan measurable.
    [Show full text]
  • LECTURE NOTES, Part I
    Prof. A. Sapozhnikov Mathematics 3 (10-PHY-BIPMA3) LECTURE NOTES, Part I Literature 1. H. Fischer, H. Kaul, Mathematik f¨urPhysiker Vol 2, 2011. 2. L. D. Kudryavtsev, A course in mathematical analysis. Vol. 2, 1988. 3. A. Sch¨uler, Calculus, Leipzig University, Lecture notes, 2006. 4. T. Tao, An introduction to measure theory, 2011, p.2{14. Contents 1 Jordan measure 2 1.1 Notation . .2 1.2 Inner and outer Jordan measures . .2 1.3 Jordan measurable sets . .5 2 Multiple integral 7 2.1 Definition and basic properties . .7 2.2 Conditions for integrability . .9 2.3 Properties of the multiple integral . 11 2.4 Iterated integrals . 15 2.5 Changes of variables . 22 2.6 Improper integrals . 25 3 Line integrals 26 3.1 Line integral of scalar field . 26 3.2 Line integral of vector field . 28 3.3 Green's formula . 30 3.3.1 Area of a set bounded by a curve . 32 3.3.2 Sign of Jacobian . 32 3.4 Conservative vector fields . 33 4 Surface integrals 35 4.1 Surfaces . 35 4.2 First fundamental form . 37 4.2.1 Length of curve . 38 4.2.2 Surface area . 38 4.2.3 Examples . 39 4.3 Surface integral of a scalar field . 40 4.4 Surface integral of a vector field . 42 4.5 Divergence (Gauss-Ostrogradsky) theorem . 44 4.6 Stokes' theorem . 46 4.7 Solenoidal vector fields . 49 1 Jordan measure 1.1 Notation We consider the Eucliean space Rn. Its elements are n-tuples of real numbers denoted by x = (x1; : : : ; xn); y = (y1; : : : ; yn);::: n p 2 2 • For x 2 R , we denote by kxk = x1 + ::: + xn the Euclidean norm of x.
    [Show full text]
  • Lecture Notes of Functional Analysis - Part 1 Degree Course: Master’S Program in Mathematics Teacher: Sisto Baldo
    Lecture Notes of Functional Analysis - Part 1 Degree Course: Master's Program in Mathematics Teacher: Sisto Baldo These notes are just a fairly detailed summary of what went on in class. In no way they are meant as a replacement for actual classes, human in- teraction with the teacher, and/or the reading of reference texts, You are of course strongly encouraged to take advantage of ALL these different learning opportunities. 1 Contents 1 Lecture of october 1, 2013 (1 hour) 5 Lebesgue measure: motivations, a brief survey on Peano-Jordan measures, outer Lebesgue measure and its elementary properties. Abstract outer mea- sures. 2 Lecture of october 2, 2013 (2 hours) 9 Outer measures, measurable sets in the sense of Caratheodory, property of the measure on measurable sets 3 Lecture of october 3, 2013 (2 hours) 13 Regularity of Lebesgue measure. Existence of sets which are not Lebesgue- measurable. Measurable functions. Simple function. Integral convergence theorems. 4 Lecture of october 7, 2013 (2 hours) 24 Comparison between Lebesgue and Riemann integrals. Fubini Theorem. 5 Lecture of october 8, 2013 (2 hours) 29 Normed spaces. Examples. Infinite sums and integration w.r.t. the counting measure. 6 Lecture of october 9, 2013 (2 hours) 32 Further examples of normed spaces. Completeness and Banach spaces. Equiv- alent and inequivalent norms. Infinite dimension: algebraic and topological dual space. Characterization of continuous linear functionals. 7 Lecture of october 10, 2013 (2 hours) 36 Normed spaces of finite dimension. Dual norm, completeness of the dual. Hahn-Banach theorem. 8 Lecture of october 14, 2013 (2 hours) 39 Proof of the Hahn-Banach theorem.
    [Show full text]
  • Partitions of Sets and the Riemann Integral
    Mathematical Communications 11(2006), 55-61 55 Partitions of sets and the Riemann integral Simeˇ Ungar∗ Abstract. We will discuss the definition of the Riemann integral using general partitions and give an elementary explication, without re- sorting to nets, generalized sequences and such, of what is meant by saying that “the Riemann integral is the limit of Darboux sums when the mesh of the partition approaches zero”. Key words: Riemann integral, Jordan measure, limit over parti- tions AMS subject classifications: 26B15, 28A75 Received January 24, 2006 Accepted June 13, 2006 1. Introduction To define the Riemann integral one traditionally starts with a bounded real-valued function f : I → R,whereI ∈ Rn is an n-dimensional parallelepiped, and usingthe Darboux sums correspondingto partitions of I, one defines Riemann integrability and the Riemann integral for such a function (e.g. [Spivak]). Next, one defines the Jordan measurable subsets of Rn as those bounded sets S for which the character- istic function is Riemann integrable on some parallelepiped I containing S.Now one defines the Riemann integral for a bounded real-valued function defined on a Jordan measurable subsets S ⊆ Rn, by extendingthe given function f over some parallelepiped I containing S by settingit equal to zero outside S, and takingits integral over I to be the integral of f over S. Without usingany measure theory, one can define what is meant by a set of (Lebesgue’s) measure zero, and prove the Lebesgue’s criterion for Riemann integrability, acquiring a powerful tool for proving basic properties of the Riemann integral as well as for recognizing Jordan measur- able sets and Riemann integrable functions defined on such sets.
    [Show full text]
  • Hewitt and Lebesgue Decompositions of States on Orthomodular Posets1
    Journal of Mathematical Analysis and Applications 255, 74–104 (2001) doi:10.1006/jmaa.2000.7160, available online at http://www.idealibrary.com on Yosida–Hewitt and Lebesgue Decompositions of States on Orthomodular Posets1 Anna De Simone Dip. di Matematica e Applicazioni “R. Caccioppoli,” Universita` degli Studi di Napoli “Federico II,” Complesso Monte S. Angelo, Via Cintia, 80126 Napoli, Italy E-mail: [email protected] and Mirko Navara Center for Machine Perception, Faculty of Electrical Engineering, Czech Technical University, Technicka´ 2, 166 27 Praha, Czech Republic E-mail: [email protected] Submitted by A. Di Nola Received December 13, 1999 Orthomodular posets are usually used as event structures of quantum mechanical systems. The states of the systems are described by probability measures (also called states) on it. It is well known that the family of all states on an orthomodular poset is a convex set, compact with respect to the product topology. This suggests using geometrical results to study its structure. In this line, we deal with the problem of the decomposition of states on orthomodular posets with respect to a given face of the state space. For particular choices of this face, we obtain, e.g., Lebesgue- type and Yosida–Hewitt decompositions as special cases. Considering, in particular, the problem of existence and uniqueness of such decompositions, we generalize to this setting numerous results obtained earlier only for orthomodular lattices and orthocomplete orthomodular posets. © 2001 Academic Press Key Words: face of a convex set; state; probability measure; orthomodular poset; Yosida–Hewitt decomposition; Lebesgue decomposition; filtering set; filtering function; heredity.
    [Show full text]
  • Graph Hausdorff Dimension, Kolmogorov Complexity And
    Graph Hausdorff dimension, Kolmogorov complexity and construction of fractal networks Leonid Bunimovich1 and Pavel Skums2,3 1School of Mathematics, Georgia Institute of Technology, 686 Cherry St NW, Atlanta, GA, USA, 30313 2Department of Computer Science, Georgia State University, 1 Park Pl NE, Atlanta, GA, USA, 30303 3Corresponding author. Email: [email protected] March 22, 2019 Abstract In this paper we introduce and study discrete analogues of Lebesgue and Hausdorff dimensions for graphs. It turned out that they are closely related to well-known graph characteristics such as rank dimension and Prague (or Neˇsetˇril-R¨odl)dimension. It allows us to formally define fractal graphs and establish fractality of some graph classes. We show, how Hausdorff dimension of graphs is related to their Kolmogorov complexity. We also demonstrate applications of this approach by establishing a novel property of general compact metric spaces using ideas from hypergraphs theory and by proving an estimation for Prague dimension of almost all graphs using methods from algorithmic information theory. Keywords: Prague dimension, rank dimension, Lebesgue dimension, Hausdorff dimension, Kolmogorov complexity, fractal 1 Introduction Lately there was a growing interest in studying self-similarity and fractal prop- erties of graphs, which is largely inspired by applications in biology, sociology and chemistry [21, 19]. Such studies often employ statistical physics methods that use ideas from graph theory and general topology, but are not intended arXiv:1607.04703v3 [math.CO] 21 Mar 2019 to approach the problems under consideration in a rigorous mathematical way. Several studies that translate certain notions of topological dimension theory to graphs using combinatorial methods are also known [20, 10].
    [Show full text]
  • Jordan Fields and Improper Integrals
    JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 133, 163-194 (1988) Jordan Fields and Improper Integrals DOROTHY MAHARAM Depurtment of Mathematics, University of Rochester, Rochester New York, 14627 Submitted by R. P. Boas Received February 19, 1987 An integral is defined for not necessarily bounded functions, with respect to a finitely additive, possibly infinite measure. In particular this strictly includes the familiar case of absolutely convergent improper Riemann integration in Euclidean space. However, the measure need not have a countably additive extension, and no topology is assumed on the underlying space. The “Jordan field,” of sets with integrable characteristic functions, is used to characterize the integrable functions as those that are “nearly” Jordan measurable, and their integrals are shown to equal the (Lebesgue) integrals of the measures of the appropriate Jordan sets. 0 1988 Academic Press. Inc 1. INTRODUCTION The motivation for this paper arose from problems in ergodic theory. There one encounters relative asymptotic densities of sequencesof integers, which give rise to (non-negative) measures (on sets of integers) that are only finitely additive and that may become infinite. The functions that one wishes to integrate with respect to such measures are in general unboun- ded. Thus it becomes desirable to develop a theory of what might be called “improper Riemann integration, ” in a reasonably general setting; that is the object of this paper. The main theorem (5.4 below) will be applied to ergodic theory in a subsequent paper. In the case of “proper Riemann integration,” of bounded functions with respect to a finite, finitely additive measure, Ridder [ll, 123 showed that the measure p extends naturally to the (finitely additive) field of sets with integrable characteristic functions--called the “Jordan field” in the present paper (since in the classical case of Riemann integration in Euclidean space it consists of the Jordan measurable sets).
    [Show full text]
  • 2 the Jordan Measure
    Physics 129a Measure Theory 071126 Frank Porter 1 Introduction The rigorous mathematical underpinning of much of what we have discussed, such as the construction of a proper Hilbert space of functions, is to be found in “measure theory”. We thus refine the concepts in our note on Hilbert spaces here. It should immediately be stated that the term “measure” refers to the notion of measuring the “size” of a set. This is the subject of measure theory. With measure theory, we will find that we can generalize the Riemann notion of an integral. [This note contains all the essential ideas to complete the development begun in the note on Hilbert spaces for the construction of a suitable Hilbert space for quantum mechanics. However, this note is still under consruction, as there remain gaps in the presentation.] Let us motivate the discussion by considering the space, C2(−1, 1) of complex-valued continuous functions on [−1, 1]. We define the norm, for any f(x) ∈ C2[−1, 1]: 1 ||f||2 = |f(x)|2dx. (1) −1 Consider the following sequence of functions, f1,f2,...,inC2[−1, 1]: ⎧ ⎨⎪ −1 −1 ≤ x ≤−1/n, fn(x)= nt −1/n ≤ x ≤ 1/n, (2) ⎩⎪ 11/n ≤ x ≤ 1. Fig. 1 illustrates the first few of these functions. This set of functions defines a Cauchy sequence, with convergence to the discontinuous function −1 −1 ≤ x ≤ 0, f(x)= (3) 10<x≤ 1. Since f(x) ∈/ C2[−1, 1], this space is not complete. How can we “complete” such a space? We need to add discontinuous functions somehow.
    [Show full text]
  • FUNDAMENTALS of REAL ANALYSIS by Do˘Gan C¸Ömez II
    FUNDAMENTALS OF REAL ANALYSIS by Do˘ganC¸¨omez II. MEASURES AND MEASURE SPACES II.1. Prelude Recall that the Riemann integral of a real-valued function f on an interval [a; b] is defined as n Z b X f(x)dx := lim f(xi)(xi − xi−1); (if the limit exists) kPk!0 a i=1 where P = fa = x1 < x2 < ··· < xn−1 < xn = bg is a partitioning of [a; b]; kPk = maxi jxi − xi−1j; and xi 2 [xi−1; xi] is arbitrary. Then, it follows that if f :[a; b] ! R is a continuous function, then it is Riemann-integrable. Also, if f :[a; b] ! R has finitely many bounded jump discontinuities, then it is Riemann-integrable. Indeed, there are functions with infinitely many bounded jumps which are still Riemann-integrable. However, there are \very simple" functions with countably many bounded jumps which are not Riemann-integrable. For example, let 1 if x 2 [0; 1] \ Q f(x) = c 0 if x 2 [0; 1] \ Q : Pn Pn Then, lim supkPk!0 i=1 f(xi)(xi − xi−1) = 1 6= 0 = lim infkPk!0 i=1 f(xi)(xi − xi−1)! Riemann integral also suffers from a convergence problem: let frig be an enumeration (listing) of rational numbers in [0; 1]; and define a sequence of functions ffng on [0; 1] by 1 if x = r1; r2; r3; : : : ; rn f (x) = n 0 otherwise; and f be the function in the example above. Then each fn is Riemann-integrable, fn ! f pointwise on [0; 1]; but f is not Riemann-integrable.
    [Show full text]
  • Uniform Distribution Theory 4 (2009), No.1, 117–145 Distribution Theory
    uniform Uniform Distribution Theory 4 (2009), no.1, 117–145 distribution theory UNIFORM DISTRIBUTION PRESERVING MAPPINGS AND VARIATIONAL PROBLEMS Stefan Steinerberger ABSTRACT. This paper studies two aspects of uniform-distribution preserving functions. We give a general criterion under which a function of this type can be expected to preserve discrepancy and apply it to study linear mappings and their effect on the isotropic discrepancy. Also, we investigate certain types of variational problems for these functions and establish some connections to uni- formly distributed sequences which will then allow us to solve similar variational problems for uniformly distributed sequences by means of these functions and their properties. Communicated by Reinhard Winkler 1. Introduction Uniform-distribution preserving functions have been independently discovered by Bosch [2] and Porubsk´y, Sal´atˇ & Strauch [17]. The theory was later extended to compact metric spaces by Tichy & Winkler [19]. Several papers dealing with similar results are given in [14]. A function : [0, 1] [0, 1] is said to be uniform distribution preserving → (henceforth: u.d.p.) if for every equidistributed sequence (xn) its image (f(xn)) is equidistributed as well. Perhaps the most fundamental property of such func- tions (and even equivalent to the given definition, see also [17]) is that for each Riemann integrable function f and each u.d.p. function the composition f is also Riemann integrable and ∘ 1 1 f((x))dx = f(x)dx. Z0 Z0 2000 Mathematics Subject Classification: Primary: 11J71, 11K06, 11K38. Sec- ondary: 41A45, 49J30. K e y w o r d s: Uniform distribution-preserving, discrepancy, box splines, variational problems.
    [Show full text]
  • Basic Lebesgue Measure Theory1
    John Nachbar Washington University in St. Louis January 10, 2018 Basic Lebesgue Measure Theory1 1 Introduction. This is an introduction to Measure Theory. I focus on the motivation for and the definition of Lebesgue measure on [0; 1]; separate notes provide a brief introduction to Lebesgue integration. My treatment largely follows Royden(2010), but I also draw on Kolmogorov and Fomin(1970), Stein and Shakarchi(2005), and Tao(2011), among others. Although Measure Theory has a deserved reputation for being subtle, its under- lying motivation is straightforward. Lebesgue measure can by viewed as a natural generalization of length to sets that are more complicated than intervals or finite unions of intervals. Viewed as a probability, Lebesgue measure is the appropriate generalization of equally likely, as captured by the uniform distribution, to compli- cated events. As an illustration of what is at issue, Section3 examines the Law of Large Numbers for tosses of a fair coin. Recall that the Strong Law of Large Numbers says that, with probability one, the empirical frequency of heads converges to 1/2. In contrast, the Weak Law of Large Numbers says, roughly, that with high probability the empirical frequency is close to 1/2. The Strong Law implies the Weak Law but not conversely. The Strong Law uses measure theoretic ideas. To understand why, note that in order to state that, \with probability one, the empirical frequency of heads converges to 1/2," one has to specify exactly what set has probability one. In the present context, the set of outcomes is the set of all possible realizations of coin tosses, which are infinite strings of the form HTTHHH.
    [Show full text]