A Brief Introduction to Lebesgue Theory
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Geometric Integration Theory Contents
Steven G. Krantz Harold R. Parks Geometric Integration Theory Contents Preface v 1 Basics 1 1.1 Smooth Functions . 1 1.2Measures.............................. 6 1.2.1 Lebesgue Measure . 11 1.3Integration............................. 14 1.3.1 Measurable Functions . 14 1.3.2 The Integral . 17 1.3.3 Lebesgue Spaces . 23 1.3.4 Product Measures and the Fubini–Tonelli Theorem . 25 1.4 The Exterior Algebra . 27 1.5 The Hausdorff Distance and Steiner Symmetrization . 30 1.6 Borel and Suslin Sets . 41 2 Carath´eodory’s Construction and Lower-Dimensional Mea- sures 53 2.1 The Basic Definition . 53 2.1.1 Hausdorff Measure and Spherical Measure . 55 2.1.2 A Measure Based on Parallelepipeds . 57 2.1.3 Projections and Convexity . 57 2.1.4 Other Geometric Measures . 59 2.1.5 Summary . 61 2.2 The Densities of a Measure . 64 2.3 A One-Dimensional Example . 66 2.4 Carath´eodory’s Construction and Mappings . 67 2.5 The Concept of Hausdorff Dimension . 70 2.6 Some Cantor Set Examples . 73 i ii CONTENTS 2.6.1 Basic Examples . 73 2.6.2 Some Generalized Cantor Sets . 76 2.6.3 Cantor Sets in Higher Dimensions . 78 3 Invariant Measures and the Construction of Haar Measure 81 3.1 The Fundamental Theorem . 82 3.2 Haar Measure for the Orthogonal Group and the Grassmanian 90 3.2.1 Remarks on the Manifold Structure of G(N,M).... 94 4 Covering Theorems and the Differentiation of Integrals 97 4.1 Wiener’s Covering Lemma and its Variants . -
The Fundamental Theorem of Calculus for Lebesgue Integral
Divulgaciones Matem´aticasVol. 8 No. 1 (2000), pp. 75{85 The Fundamental Theorem of Calculus for Lebesgue Integral El Teorema Fundamental del C´alculo para la Integral de Lebesgue Di´omedesB´arcenas([email protected]) Departamento de Matem´aticas.Facultad de Ciencias. Universidad de los Andes. M´erida.Venezuela. Abstract In this paper we prove the Theorem announced in the title with- out using Vitali's Covering Lemma and have as a consequence of this approach the equivalence of this theorem with that which states that absolutely continuous functions with zero derivative almost everywhere are constant. We also prove that the decomposition of a bounded vari- ation function is unique up to a constant. Key words and phrases: Radon-Nikodym Theorem, Fundamental Theorem of Calculus, Vitali's covering Lemma. Resumen En este art´ıculose demuestra el Teorema Fundamental del C´alculo para la integral de Lebesgue sin usar el Lema del cubrimiento de Vi- tali, obteni´endosecomo consecuencia que dicho teorema es equivalente al que afirma que toda funci´onabsolutamente continua con derivada igual a cero en casi todo punto es constante. Tambi´ense prueba que la descomposici´onde una funci´onde variaci´onacotada es ´unicaa menos de una constante. Palabras y frases clave: Teorema de Radon-Nikodym, Teorema Fun- damental del C´alculo,Lema del cubrimiento de Vitali. Received: 1999/08/18. Revised: 2000/02/24. Accepted: 2000/03/01. MSC (1991): 26A24, 28A15. Supported by C.D.C.H.T-U.L.A under project C-840-97. 76 Di´omedesB´arcenas 1 Introduction The Fundamental Theorem of Calculus for Lebesgue Integral states that: A function f :[a; b] R is absolutely continuous if and only if it is ! 1 differentiable almost everywhere, its derivative f 0 L [a; b] and, for each t [a; b], 2 2 t f(t) = f(a) + f 0(s)ds: Za This theorem is extremely important in Lebesgue integration Theory and several ways of proving it are found in classical Real Analysis. -
[Math.FA] 3 Dec 1999 Rnfrneter for Theory Transference Introduction 1 Sas Ihnrah U Twl Etetdi Eaaepaper
Transference in Spaces of Measures Nakhl´eH. Asmar,∗ Stephen J. Montgomery–Smith,† and Sadahiro Saeki‡ 1 Introduction Transference theory for Lp spaces is a powerful tool with many fruitful applications to sin- gular integrals, ergodic theory, and spectral theory of operators [4, 5]. These methods afford a unified approach to many problems in diverse areas, which before were proved by a variety of methods. The purpose of this paper is to bring about a similar approach to spaces of measures. Our main transference result is motivated by the extensions of the classical F.&M. Riesz Theorem due to Bochner [3], Helson-Lowdenslager [10, 11], de Leeuw-Glicksberg [6], Forelli [9], and others. It might seem that these extensions should all be obtainable via transference methods, and indeed, as we will show, these are exemplary illustrations of the scope of our main result. It is not straightforward to extend the classical transference methods of Calder´on, Coif- man and Weiss to spaces of measures. First, their methods make use of averaging techniques and the amenability of the group of representations. The averaging techniques simply do not work with measures, and do not preserve analyticity. Secondly, and most importantly, their techniques require that the representation is strongly continuous. For spaces of mea- sures, this last requirement would be prohibitive, even for the simplest representations such as translations. Instead, we will introduce a much weaker requirement, which we will call ‘sup path attaining’. By working with sup path attaining representations, we are able to prove a new transference principle with interesting applications. For example, we will show how to derive with ease generalizations of Bochner’s theorem and Forelli’s main result. -
Some Inequalities in the Theory of Functions^)
SOME INEQUALITIES IN THE THEORY OF FUNCTIONS^) BY ZEEV NEHARI 1. Introduction. Many of the inequalities of function theory and potential theory may be reduced to statements regarding the properties of harmonic domain functions with vanishing or constant boundary values, that is, func- tions which can be obtained from the Green's function by means of elementary processes. For the derivation of these inequalities a large number of different techniques and procedures have been used. It is the aim of this paper to show that many of the known inequalities of this type, and also others which are new, can be obtained as simple consequences of the classical minimum prop- erty of the Dirichlet integral. In addition to the resulting simplification, this method has the further advantage of being capable of generalization to a wide class of linear partial differential equations of elliptic type in two or more variables. The idea of using the positive-definite character of an integral as the point of departure for the derivation of function-theoretic inequalities is, of course, not new and it has been successfully used for this purpose by a num- ber of authors [l; 2; 8; 9; 16]. What the present paper attempts is to give a more or less systematic survey of the type of inequality obtainable in this way. 2. Monotonie functionals. 1. The domains we shall consider will be as- sumed to be bounded by a finite number of closed analytic curves and they will be embedded in a given closed Riemann surface R of finite genus. The symbol 5(a) will be used to denote a "singularity function" with the follow- ing properties: 5(a) is real, harmonic, and single-valued on R, with the excep- tion of a finite number of points at which 5(a) has specified singularities. -
Generalizations of the Riemann Integral: an Investigation of the Henstock Integral
Generalizations of the Riemann Integral: An Investigation of the Henstock Integral Jonathan Wells May 15, 2011 Abstract The Henstock integral, a generalization of the Riemann integral that makes use of the δ-fine tagged partition, is studied. We first consider Lebesgue’s Criterion for Riemann Integrability, which states that a func- tion is Riemann integrable if and only if it is bounded and continuous almost everywhere, before investigating several theoretical shortcomings of the Riemann integral. Despite the inverse relationship between integra- tion and differentiation given by the Fundamental Theorem of Calculus, we find that not every derivative is Riemann integrable. We also find that the strong condition of uniform convergence must be applied to guarantee that the limit of a sequence of Riemann integrable functions remains in- tegrable. However, by slightly altering the way that tagged partitions are formed, we are able to construct a definition for the integral that allows for the integration of a much wider class of functions. We investigate sev- eral properties of this generalized Riemann integral. We also demonstrate that every derivative is Henstock integrable, and that the much looser requirements of the Monotone Convergence Theorem guarantee that the limit of a sequence of Henstock integrable functions is integrable. This paper is written without the use of Lebesgue measure theory. Acknowledgements I would like to thank Professor Patrick Keef and Professor Russell Gordon for their advice and guidance through this project. I would also like to acknowledge Kathryn Barich and Kailey Bolles for their assistance in the editing process. Introduction As the workhorse of modern analysis, the integral is without question one of the most familiar pieces of the calculus sequence. -
Stability in the Almost Everywhere Sense: a Linear Transfer Operator Approach ∗ R
CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector J. Math. Anal. Appl. 368 (2010) 144–156 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa Stability in the almost everywhere sense: A linear transfer operator approach ∗ R. Rajaram a, ,U.Vaidyab,M.Fardadc, B. Ganapathysubramanian d a Dept. of Math. Sci., 3300, Lake Rd West, Kent State University, Ashtabula, OH 44004, United States b Dept. of Elec. and Comp. Engineering, Iowa State University, Ames, IA 50011, United States c Dept. of Elec. Engineering and Comp. Sci., Syracuse University, Syracuse, NY 13244, United States d Dept. of Mechanical Engineering, Iowa State University, Ames, IA 50011, United States article info abstract Article history: The problem of almost everywhere stability of a nonlinear autonomous ordinary differential Received 20 April 2009 equation is studied using a linear transfer operator framework. The infinitesimal generator Available online 20 February 2010 of a linear transfer operator (Perron–Frobenius) is used to provide stability conditions of Submitted by H. Zwart an autonomous ordinary differential equation. It is shown that almost everywhere uniform stability of a nonlinear differential equation, is equivalent to the existence of a non-negative Keywords: Almost everywhere stability solution for a steady state advection type linear partial differential equation. We refer Advection equation to this non-negative solution, verifying almost everywhere global stability, as Lyapunov Density function density. A numerical method using finite element techniques is used for the computation of Lyapunov density. © 2010 Elsevier Inc. All rights reserved. 1. Introduction Stability analysis of an ordinary differential equation is one of the most fundamental problems in the theory of dynami- cal systems. -
Chapter 6. Integration §1. Integrals of Nonnegative Functions Let (X, S, Μ
Chapter 6. Integration §1. Integrals of Nonnegative Functions Let (X, S, µ) be a measure space. We denote by L+ the set of all measurable functions from X to [0, ∞]. + Let φ be a simple function in L . Suppose f(X) = {a1, . , am}. Then φ has the Pm −1 standard representation φ = j=1 ajχEj , where Ej := f ({aj}), j = 1, . , m. We define the integral of φ with respect to µ to be m Z X φ dµ := ajµ(Ej). X j=1 Pm For c ≥ 0, we have cφ = j=1(caj)χEj . It follows that m Z X Z cφ dµ = (caj)µ(Ej) = c φ dµ. X j=1 X Pm Suppose that φ = j=1 ajχEj , where aj ≥ 0 for j = 1, . , m and E1,...,Em are m Pn mutually disjoint measurable sets such that ∪j=1Ej = X. Suppose that ψ = k=1 bkχFk , where bk ≥ 0 for k = 1, . , n and F1,...,Fn are mutually disjoint measurable sets such n that ∪k=1Fk = X. Then we have m n n m X X X X φ = ajχEj ∩Fk and ψ = bkχFk∩Ej . j=1 k=1 k=1 j=1 If φ ≤ ψ, then aj ≤ bk, provided Ej ∩ Fk 6= ∅. Consequently, m m n m n n X X X X X X ajµ(Ej) = ajµ(Ej ∩ Fk) ≤ bkµ(Ej ∩ Fk) = bkµ(Fk). j=1 j=1 k=1 j=1 k=1 k=1 In particular, if φ = ψ, then m n X X ajµ(Ej) = bkµ(Fk). j=1 k=1 In general, if φ ≤ ψ, then m n Z X X Z φ dµ = ajµ(Ej) ≤ bkµ(Fk) = ψ dµ. -
2. Convergence Theorems
2. Convergence theorems In this section we analyze the dynamics of integrabilty in the case when se- quences of measurable functions are considered. Roughly speaking, a “convergence theorem” states that integrability is preserved under taking limits. In other words, ∞ if one has a sequence (fn)n=1 of integrable functions, and if f is some kind of a limit of the fn’s, then we would like to conclude that f itself is integrable, as well R R as the equality f = limn→∞ fn. Such results are often employed in two instances: A. When we want to prove that some function f is integrable. In this case ∞ we would look for a sequence (fn)n=1 of integrable approximants for f. B. When we want to construct and integrable function. In this case, we will produce first the approximants, and then we will examine the existence of the limit. The first convergence result, which is somehow primite, but very useful, is the following. Lemma 2.1. Let (X, A, µ) be a finite measure space, let a ∈ (0, ∞) and let fn : X → [0, a], n ≥ 1, be a sequence of measurable functions satisfying (a) f1 ≥ f2 ≥ · · · ≥ 0; (b) limn→∞ fn(x) = 0, ∀ x ∈ X. Then one has the equality Z (1) lim fn dµ = 0. n→∞ X Proof. Let us define, for each ε > 0, and each integer n ≥ 1, the set ε An = {x ∈ X : fn(x) ≥ ε}. ε Obviously, we have An ∈ A, ∀ ε > 0, n ≥ 1. One key fact we are going to use is the following. -
Lecture 26: Dominated Convergence Theorem
Lecture 26: Dominated Convergence Theorem Continuation of Fatou's Lemma. + + Corollary 0.1. If f 2 L and ffn 2 L : n 2 Ng is any sequence of functions such that fn ! f almost everywhere, then Z Z f 6 lim inf fn: X X Proof. Let fn ! f everywhere in X. That is, lim inf fn(x) = f(x) (= lim sup fn also) for all x 2 X. Then, by Fatou's lemma, Z Z Z f = lim inf fn 6 lim inf fn: X X X If fn 9 f everywhere in X, then let E = fx 2 X : lim inf fn(x) 6= f(x)g. Since fn ! f almost everywhere in X, µ(E) = 0 and Z Z Z Z f = f and fn = fn 8n X X−E X X−E thus making fn 9 f everywhere in X − E. Hence, Z Z Z Z f = f 6 lim inf fn = lim inf fn: X X−E X−E X χ Example 0.2 (Strict inequality). Let Sn = [n; n + 1] ⊂ R and fn = Sn . Then, f = lim inf fn = 0 and Z Z 0 = f dµ < lim inf fn dµ = 1: R R Example 0.3 (Importance of non-negativity). Let Sn = [n; n + 1] ⊂ R and χ fn = − Sn . Then, f = lim inf fn = 0 but Z Z 0 = f dµ > lim inf fn dµ = −1: R R 1 + R Proposition 0.4. If f 2 L and X f dµ < 1, then (a) the set A = fx 2 X : f(x) = 1g is a null set and (b) the set B = fx 2 X : f(x) > 0g is σ-finite. -
On Liouville Theorems for Harmonic Functions with Finite Dirichlet Integral Udc 517.95
. C6opHHK Math. USSR Sbornik TOM 132(174)(1987), Ban. 4 Vol. 60(1988), No. 2 ON LIOUVILLE THEOREMS FOR HARMONIC FUNCTIONS WITH FINITE DIRICHLET INTEGRAL UDC 517.95 A. A. GRIGOR'YAN ABSTRACT. A criterion for the validity of the D-Liouville theorem is proved. In §1 it is shown that the question of L°°- and D-Liouville theorems reduces to the study of the so-called massive sets (in other words, the level sets of harmonic functions in the classes L°° and L°° Π D). In §2 some properties of capacity are presented. In §3 the criterion of D-massiveness is formulated—the central result of this article—and examples are presented. In §4 a criterion for the D-Liouville theorem is formulated, and corollaries are derived. In §§5-9 the main theorems are proved. Figures: 5. Bibliography: 17 titles. Introduction The classical theorem of Liouville states that any bounded harmonic function on Rn is constant. It is easy to verify that the following assertions are also true: 1) If the harmonic function u on R™ has finite Dirichlet integral then u = const. 2) If u € LP(R") is a harmonic function, 1 < ρ < oo, then «ΞΟ. The list of theorems of this kind can be extended; they are known in the literature under the general category of Liouville-type theorems. After Moser's paper [1], which in particular proved Liouville's theorem for entire solutions of the uniformly elliptic equation it became possible to study the solutions of the Laplace-Beltrami equation^) on arbitrary Riemannian manifolds. The main efforts here are directed towards finding under what geometric conditions one or another Liouville theorem is true. -
MEASURE and INTEGRATION: LECTURE 3 Riemann Integral. If S Is Simple and Measurable Then Sdµ = Αiµ(
MEASURE AND INTEGRATION: LECTURE 3 Riemann integral. If s is simple and measurable then N � � sdµ = αiµ(Ei), X i=1 �N where s = i=1 αiχEi . If f ≥ 0, then � �� � fdµ = sup sdµ | 0 ≤ s ≤ f, s simple & measurable . X X Recall the Riemann integral of function f on interval [a, b]. Define lower and upper integrals L(f, P ) and U(f, P ), where P is a partition of [a, b]. Set − � � f = sup L(f, P ) and f = inf U(f, P ). P P − A function f is Riemann integrable ⇐⇒ − � � f = f, − in which case this common value is � f. A set B ⊂ R has measure zero if, for any � > 0, there exists a ∞ ∞ countable collection of intervals { Ii}i =1 such that B ⊂ ∪i =1Ii and �∞ i=1 λ(Ii) < �. Examples: finite sets, countable sets. There are also uncountable sets with measure zero. However, any interval does not have measure zero. Theorem 0.1. A function f is Riemann integrable if and only if f is discontinuous on a set of measure zero. A function is said to have a property (e.g., continuous) almost ev erywhere (abbreviated a.e.) if the set on which the property does not hold has measure zero. Thus, the statement of the theorem is that f is Riemann integrable if and only if it is continuous almost everywhere. Date: September 11, 2003. 1 2 MEASURE AND INTEGRATION: LECTURE 3 Recall positive measure: a measure function µ: M → [0, ∞] such ∞ �∞ that µ (∪i=1Ei) = i=1 µ(Ei) for Ei ∈ M disjoint. -
Lecture Notes for Math 522 Spring 2012 (Rudin Chapter
Lebesgue integral We will define a very powerful integral, far better than Riemann in the sense that it will allow us to integrate pretty much every reasonable function and we will also obtain strong convergence results. That is if we take a limit of integrable functions we will get an integrable function and the limit of the integrals will be the integral of the limit under very mild conditions. We will focus only on the real line, although the theory easily extends to more abstract contexts. In Riemann integral the basic block was a rectangle. If we wanted to integrate a function that was identically 1 on an interval [a; b], then the integral was simply the area of that rectangle, so 1×(b−a) = b−a. For Lebesgue integral what we want to do is to replace the interval with a more general subset of the real line. That is, if we have a set S ⊂ R and we take the indicator function or characteristic function χS defined by ( 1 if x 2 S, χ (x) = S 0 else. Then the integral of χS should really be equal to the area under the graph, which should be equal to the \size" of S. Example: Suppose that S is the set of rational numbers between 0 and 1. Let us argue that its size is 0, and so the integral of χS should be 0. Let fx1; x2;:::g = S be an enumeration of the points of S. Now for any > 0 take the sets −j−1 −j−1 Ij = (xj − 2 ; xj + 2 ); then 1 [ S ⊂ Ij: j=1 −j The \size" of any Ij should be 2 , so it seems reasonable to say that the \size" of S is less than the sum of the sizes of the Ij's.