DOCSLIB.ORG

Measure and Integral

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Measure and Integral JUHA KINNUNEN Measure and Integral Department of Mathematics, Aalto University 2020 Contents 1 MEASURE THEORY1 1.1 Outer measures ............................... 1 1.2 Measurable sets ............................... 6 1.3 Measures ................................... 11 1.4 The distance function ............................ 17 1.5 Characterizations of measurable sets .................. 19 1.6 Metric outer measures ........................... 30 1.7 Lebesgue measure revisited ........................ 34 1.8 Invariance properties of the Lebesgue measure ............ 42 1.9 Lebesgue measurable sets ........................ 44 1.10 A nonmeasurable set ............................ 48 1.11 The Cantor set ................................ 52 2 MEASURABLE FUNCTIONS 55 2.1 Calculus with infinities ............................ 55 2.2 Measurable functions ............................ 56 2.3 Cantor-Lebesgue function ......................... 62 2.4 Lipschitz mappings on Rn ......................... 65 2.5 Limits of measurable functions ...................... 68 2.6 Almost everywhere ............................. 69 2.7 Approximation by simple functions .................... 71 2.8 Modes of convergence .......................... 74 2.9 Egoroff’s and Lusin’s theorems ...................... 79 3 INTEGRATION 85 3.1 Integral of a nonnegative simple function ............... 85 3.2 Integral of a nonnegative measurable function ............ 88 3.3 Monotone convergence theorem .................... 91 3.4 Fatou’s lemma ................................ 96 3.5 Integral of a signed function ........................ 96 3.6 Dominated convergence theorem ................... 100 3.7 Lebesgue integral .............................. 105 3.8 Cavalieri’s principle ............................. 116 CONTENTS ii 3.9 Lebesgue and Riemann .......................... 121 3.10 Fubini’s theorem ............................... 125 3.11 Fubini’s theorem for Lebesgue measure ................. 130 All sets can be measured by an outer measure, which is monotone and countably subadditive function. Most im- portant example is the Lebesgue outer measure, which generalizes the concept of volume to all sets. An outer measure has a proper measure theory on measurable sets. A set is Lebesgue measurable if it is almost a Borel set. Existence of a nonmeasurable set for the Lebesgue outer measure is shown by the axiom of choice. 1 Measure theory 1.1 Outer measures We begin with a general definition of outer measure Let X be a set and consider a mapping on the collection of subsets of X. Recall that if Ai X for i 1,2,..., ½ Æ then [1 Ai {x X : x Ai for some i} i 1 Æ 2 2 Æ and \1 Ai {x X : x Ai for every i}. i 1 Æ 2 2 Æ Moreover X \ A {x X : x A}. Æ 2 ∉ Definition 1.1. A mapping ¹¤ : {A : A X} [0, ] is an outer measure on X, if ½ ! 1 (1) ¹¤( ) 0, ; Æ (2) (monotonicity) ¹¤(A) ¹¤(B) whenever A B X and É ½ ½ ¡S ¢ P (3) (countable subadditivity) ¹¤ 1i 1 Ai 1i 1 ¹¤(Ai). Æ É Æ T HEMORAL : An outer measure is a countably subadditive set function. Countable subadditivity implies finite subdditivity, since we may add countably many empty sets. W ARNING : It may happen that the equality ¹¤(A B) ¹¤(A) ¹¤(B) fails [ Æ Å for A and B with A B . This means that an outer measure is not necessarily \ Æ; additive on pairwise disjoint sets. Observe that holds by subadditivity. É Example 1.2. Let X {1,2,3} and define ¹¤( ) 0, ¹¤(X) 2 and ¹¤(E) 1 for all Æ ; Æ Æ Æ other E X. Then ¹¤ is an outer measure on X. However, if A {1} and B {2}, ½ Æ Æ then ¹¤(A B) ¹¤({1,2}) 1 2 ¹¤(A) ¹¤(B). [ Æ Æ 6Æ Æ Å 1 CHAPTER 1. MEASURE THEORY 2 Figure 1.1: Countable subadditivity. Examples 1.3: (1) (The trivial measure) Let ¹¤(A) 0 for every A X. Then ¹¤ is an outer Æ ½ measure. The trivial measure is relatively useless, since all sets have measure zero. (2) (The discrete measure) Let 8 <1, A , ¹¤(A) 6Æ ; Æ :0, A . Æ; The discrete outer measure tells whether or not a set is empty. (3) (The Dirac measure) Let x0 X be a fixed point and let 2 8 <1, x0 A, ¹¤(A) 2 Æ :0, x0 A. ∉ This is called the Dirac outer measure at x0. The Dirac measure tells whether or not a set contains the point x0. (4) (The counting measure) Let ¹¤(A) be the (possibly infinite) number of points in A. The counting outer measure tells the number of points of a set. (5) (The Lebesgue measure) Let X Rn and consider the n-dimensional inter- Æ val n I {x R : ai xi bi, i 1,..., n} [a1, b1] ... [an, bn] Æ 2 É É Æ Æ £ £ CHAPTER 1. MEASURE THEORY 3 with sides parallel to the coordinate axes. We allow intervals to be degen- erate, that is, bi ai for some i. The geometric volume of I is Æ vol(I) (b1 a1)(b2 a2) (bn an). Æ ¡ ¡ ¢¢¢ ¡ The Lebesgue outer measure of a set A Rn is defined as ½ ( ) X1 [1 m¤(A) inf vol(Ii) : A Ii , Æ i 1 ½ i 1 Æ Æ where the infimum is taken over all coverings of A by countably many intervals Ii, i 1,2,.... Observe that this includes coverings with a finite Æ number of intervals, since we may add countably many intervals Ii {xi} Æ containing only one point with vol(Ii) 0. The Lebesgue outer measure is Æ nonnegative but may be infinite, so that 0 m¤(A) . By the definition É É 1 of infimum, for every " 0, there are intervals Ii, i 1,2,..., such that S È Æ A 1i 1 Ii and ½ Æ X1 m¤(A) vol(Ii) m¤(A) ". É i 1 Ç Å Æ T HEMORAL : The Lebesgue outer measure of a set is the infimum of sums of volumes of countably many intervals that cover the set. We shall discuss more about the Lebesgue outer measure later, but it generalizes the notion of n-dimensional volume to arbitrary subsets of Rn. Figure 1.2: Covering by intervals. CHAPTER 1. MEASURE THEORY 4 Claim: m¤ is an outer measure Reason. (1) Let " 0. Since [ "1/n/2,"1/n/2]n, we have È ; ½ ¡ 1/n 1/n n 1/n n 0 m¤( ) vol([ " /2," /2] ) (2" /2) ". É ; É ¡ Æ Æ By letting " 0, we conclude m¤( ) 0. We could also cover by the ! ; Æ ; n degenerate interval [x1, x1] [xn, xn] for any x (x1,..., xn) R and £ ¢¢¢ £ Æ 2 conclude the claim from this. (2) Let A B. We may assume that m¤(B) , for otherwise m¤(A) ½ Ç 1 É m¤(B) and there is nothing to prove. For every " 0 there are intervals Æ 1 S È Ii, i 1,2,..., such that B 1i 1 Ii and Æ ½ Æ X1 vol(Ii) m¤(B) ". i 1 Ç Å Æ S Since A B 1i 1 Ii, we have ½ ½ Æ X1 m¤(A) vol(Ii) m¤(B) ". É i 1 Ç Å Æ By letting " 0, we conclude m¤(A) m¤(B). ! É (3) We may assume that m¤(Ai) for every i 1,2,..., for otherwise Ç 1 Æ there is nothing to prove. Let " 0. For every i 1,2,... there are intervals ÈS Æ I j,i, j 1,2,..., such that Ai 1j 1 I j,i and Æ ½ Æ " X1 vol(I ) m (A ) . j,i ¤ i i j 1 Ç Å 2 Æ S S S S Then 1i 1 Ai 1i 1 1j 1 I j,i 1i, j 1 I j,i and Æ ½ Æ Æ Æ Æ Ã ! [1 X1 X1 X1 m¤ Ai vol(I j,i) vol(I j,i) i 1 É i, j 1 Æ i 1 j 1 Æ Æ Æ Æ µ " ¶ X1 m (A ) X1 m (A ) . ¤ i i ¤ i " É i 1 Å 2 Æ i 1 Å Æ Æ The claim follows by letting " 0. ! ■ (6) (The Hausdorff measure) Let X Rn, 0 s and 0 ± . Define Æ Ç Ç 1 Ç É 1 ( ) s X1 s : [1 H± (A) inf (diam(Bi)) A Bi, diam(Bi) ± Æ i 1 ½ i 1 É Æ Æ and s s s H (A) lim H± (A) supH± (A). Æ ± 0 Æ ± 0 ! È We call H s the s-dimensional Hausdorff outer measure on Rn. This gen- eralizes the notion of s-dimensional measure to arbitrary subsets of Rn. CHAPTER 1. MEASURE THEORY 5 See Remark 1.23 for the definition of the diameter of a set. We refer to [2, Section 3.8], [4, Chapter 2], [8, Chapter 19], [11, Chapter 7] and [16, Chapter 7] for more. Observe that, for every ± 0, an arbitrary set A Rn can be covered by È ½ B(x, ± ) with x A, that is, 2 2 ½ µ ± ¶ ¾ A [ B x, : x A , ½ 2 2 where B(x, r) {y Rn : y x r} denotes an open ball of radius r 0 Æ 2 j ¡ j Ç È and center x. By Lindelöf’s theorem every open covering in Rn has a countable subcovering. This implies that there exist countably many ± S balls Bi B(xi, 2 ), i 1,2,..., such that A 1i 1 Bi. Moreover, we have Æ Æ ½ Æ diam(Bi) ± for every i 1,2,.... This shows that the coverings in the É Æ definition of the Hausdorff outer measure exist. (7) Let F be a collection of subsets of X such that F and there exist ; 2 S : Ai F , i 1,2,..., such that X 1i 1 Ai. Let ½ F [0, ] be any 2 Æ Æ Æ ! 1 function for which ½( ) 0. Then ¹¤ : P (X) [0, ], ; Æ ! 1 ( ) X1 [1 ¹¤(A) inf ½(Ai) : Ai F , A Ai Æ i 1 2 ½ i 1 Æ Æ is an outer measure on X. Moreover, if ½ is monotone and countably subadditive on F , then ¹¤ ½ on F (exercise).
Load more

Recommended publications
  • Measure Theory (Graduate Texts in Mathematics)
    PaulR. Halmos Measure Theory Springer-VerlagNewYork-Heidelberg-Berlin Managing Editors P. R. Halmos C. C. Moore Indiana University University of California Department of Mathematics at Berkeley Swain Hall East Department of Mathematics Bloomington, Indiana 47401 Berkeley, California 94720 AMS Subject Classifications (1970) Primary: 28 - 02, 28A10, 28A15, 28A20, 28A25, 28A30, 28A35, 28A40, 28A60, 28A65, 28A70 Secondary: 60A05, 60Bxx Library of Congress Cataloging in Publication Data Halmos, Paul Richard, 1914- Measure theory. (Graduate texts in mathematics, 18) Reprint of the ed. published by Van Nostrand, New York, in series: The University series in higher mathematics. Bibliography: p. 1. Measure theory. I. Title. II. Series. [QA312.H26 1974] 515'.42 74-10690 ISBN 0-387-90088-8 All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag. © 1950 by Litton Educational Publishing, Inc. and 1974 by Springer-Verlag New York Inc. Printed in the United States of America. ISBN 0-387-90088-8 Springer-Verlag New York Heidelberg Berlin ISBN 3-540-90088-8 Springer-Verlag Berlin Heidelberg New York PREFACE My main purpose in this book is to present a unified treatment of that part of measure theory which in recent years has shown itself to be most useful for its applications in modern analysis. If I have accomplished my purpose, then the book should be found usable both as a text for students and as a source of refer­ ence for the more advanced mathematician. I have tried to keep to a minimum the amount of new and unusual terminology and notation.
    [Show full text]
  • A Convenient Category for Higher-Order Probability Theory
    A Convenient Category for Higher-Order Probability Theory Chris Heunen Ohad Kammar Sam Staton Hongseok Yang University of Edinburgh, UK University of Oxford, UK University of Oxford, UK University of Oxford, UK Abstract—Higher-order probabilistic programming languages 1 (defquery Bayesian-linear-regression allow programmers to write sophisticated models in machine 2 let let sample normal learning and statistics in a succinct and structured way, but step ( [f ( [s ( ( 0.0 3.0)) 3 sample normal outside the standard measure-theoretic formalization of proba- b ( ( 0.0 3.0))] 4 fn + * bility theory. Programs may use both higher-order functions and ( [x] ( ( s x) b)))] continuous distributions, or even define a probability distribution 5 (observe (normal (f 1.0) 0.5) 2.5) on functions. But standard probability theory does not handle 6 (observe (normal (f 2.0) 0.5) 3.8) higher-order functions well: the category of measurable spaces 7 (observe (normal (f 3.0) 0.5) 4.5) is not cartesian closed. 8 (observe (normal (f 4.0) 0.5) 6.2) Here we introduce quasi-Borel spaces. We show that these 9 (observe (normal (f 5.0) 0.5) 8.0) spaces: form a new formalization of probability theory replacing 10 (predict :f f))) measurable spaces; form a cartesian closed category and so support higher-order functions; form a well-pointed category and so support good proof principles for equational reasoning; and support continuous probability distributions. We demonstrate the use of quasi-Borel spaces for higher-order functions and proba- bility by: showing that a well-known construction of probability theory involving random functions gains a cleaner expression; and generalizing de Finetti’s theorem, that is a crucial theorem in probability theory, to quasi-Borel spaces.
    [Show full text]
  • Jointly Measurable and Progressively Measurable Stochastic Processes
    Jointly measurable and progressively measurable stochastic processes Jordan Bell [email protected] Department of Mathematics, University of Toronto June 18, 2015 1 Jointly measurable stochastic processes d Let E = R with Borel E , let I = R≥0, which is a topological space with the subspace topology inherited from R, and let (Ω; F ;P ) be a probability space. For a stochastic process (Xt)t2I with state space E, we say that X is jointly measurable if the map (t; !) 7! Xt(!) is measurable BI ⊗ F ! E . For ! 2 Ω, the path t 7! Xt(!) is called left-continuous if for each t 2 I, Xs(!) ! Xt(!); s " t: We prove that if the paths of a stochastic process are left-continuous then the stochastic process is jointly measurable.1 Theorem 1. If X is a stochastic process with state space E and all the paths of X are left-continuous, then X is jointly measurable. Proof. For n ≥ 1, t 2 I, and ! 2 Ω, let 1 n X Xt (!) = 1[k2−n;(k+1)2−n)(t)Xk2−n (!): k=0 n Each X is measurable BI ⊗ F ! E : for B 2 E , 1 n [ −n −n f(t; !) 2 I × Ω: Xt (!) 2 Bg = [k2 ; (k + 1)2 ) × fXk2−n 2 Bg: k=0 −n −n Let t 2 I. For each n there is a unique kn for which t 2 [kn2 ; (kn +1)2 ), and n −n −n thus Xt (!) = Xkn2 (!). Furthermore, kn2 " t, and because s 7! Xs(!) is n −n left-continuous, Xkn2 (!) ! Xt(!).
    [Show full text]
  • Shape Analysis, Lebesgue Integration and Absolute Continuity Connections
    NISTIR 8217 Shape Analysis, Lebesgue Integration and Absolute Continuity Connections Javier Bernal This publication is available free of charge from: https://doi.org/10.6028/NIST.IR.8217 NISTIR 8217 Shape Analysis, Lebesgue Integration and Absolute Continuity Connections Javier Bernal Applied and Computational Mathematics Division Information Technology Laboratory This publication is available free of charge from: https://doi.org/10.6028/NIST.IR.8217 July 2018 INCLUDES UPDATES AS OF 07-18-2018; SEE APPENDIX U.S. Department of Commerce Wilbur L. Ross, Jr., Secretary National Institute of Standards and Technology Walter Copan, NIST Director and Undersecretary of Commerce for Standards and Technology ______________________________________________________________________________________________________ This Shape Analysis, Lebesgue Integration and publication Absolute Continuity Connections Javier Bernal is National Institute of Standards and Technology, available Gaithersburg, MD 20899, USA free of Abstract charge As shape analysis of the form presented in Srivastava and Klassen’s textbook “Functional and Shape Data Analysis” is intricately related to Lebesgue integration and absolute continuity, it is advantageous from: to have a good grasp of the latter two notions. Accordingly, in these notes we review basic concepts and results about Lebesgue integration https://doi.org/10.6028/NIST.IR.8217 and absolute continuity. In particular, we review fundamental results connecting them to each other and to the kind of shape analysis, or more generally, functional data analysis presented in the aforeme- tioned textbook, in the process shedding light on important aspects of all three notions. Many well-known results, especially most results about Lebesgue integration and some results about absolute conti- nuity, are presented without proofs.
    [Show full text]
  • LEBESGUE MEASURE and L2 SPACE. Contents 1. Measure Spaces 1 2. Lebesgue Integration 2 3. L2 Space 4 Acknowledgments 9 References
    LEBESGUE MEASURE AND L2 SPACE. ANNIE WANG Abstract. This paper begins with an introduction to measure spaces and the Lebesgue theory of measure and integration. Several important theorems regarding the Lebesgue integral are then developed. Finally, we prove the completeness of the L2(µ) space and show that it is a metric space, and a Hilbert space. Contents 1. Measure Spaces 1 2. Lebesgue Integration 2 3. L2 Space 4 Acknowledgments 9 References 9 1. Measure Spaces Definition 1.1. Suppose X is a set. Then X is said to be a measure space if there exists a σ-ring M (that is, M is a nonempty family of subsets of X closed under countable unions and under complements)of subsets of X and a non-negative countably additive set function µ (called a measure) defined on M . If X 2 M, then X is said to be a measurable space. For example, let X = Rp, M the collection of Lebesgue-measurable subsets of Rp, and µ the Lebesgue measure. Another measure space can be found by taking X to be the set of all positive integers, M the collection of all subsets of X, and µ(E) the number of elements of E. We will be interested only in a special case of the measure, the Lebesgue measure. The Lebesgue measure allows us to extend the notions of length and volume to more complicated sets. Definition 1.2. Let Rp be a p-dimensional Euclidean space . We denote an interval p of R by the set of points x = (x1; :::; xp) such that (1.3) ai ≤ xi ≤ bi (i = 1; : : : ; p) Definition 1.4.
    [Show full text]
  • 1 Measurable Functions
    36-752 Advanced Probability Overview Spring 2018 2. Measurable Functions, Random Variables, and Integration Instructor: Alessandro Rinaldo Associated reading: Sec 1.5 of Ash and Dol´eans-Dade; Sec 1.3 and 1.4 of Durrett. 1 Measurable Functions 1.1 Measurable functions Measurable functions are functions that we can integrate with respect to measures in much the same way that continuous functions can be integrated \dx". Recall that the Riemann integral of a continuous function f over a bounded interval is defined as a limit of sums of lengths of subintervals times values of f on the subintervals. The measure of a set generalizes the length while elements of the σ-field generalize the intervals. Recall that a real-valued function is continuous if and only if the inverse image of every open set is open. This generalizes to the inverse image of every measurable set being measurable. Definition 1 (Measurable Functions). Let (Ω; F) and (S; A) be measurable spaces. Let f :Ω ! S be a function that satisfies f −1(A) 2 F for each A 2 A. Then we say that f is F=A-measurable. If the σ-field’s are to be understood from context, we simply say that f is measurable. Example 2. Let F = 2Ω. Then every function from Ω to a set S is measurable no matter what A is. Example 3. Let A = f?;Sg. Then every function from a set Ω to S is measurable, no matter what F is. Proving that a function is measurable is facilitated by noticing that inverse image commutes with union, complement, and intersection.
    [Show full text]
  • Hausdorff Measure
    Hausdorff Measure Jimmy Briggs and Tim Tyree December 3, 2016 1 1 Introduction In this report, we explore the the measurement of arbitrary subsets of the metric space (X; ρ); a topological space X along with its distance function ρ. We introduce Hausdorff Measure as a natural way of assigning sizes to these sets, especially those of smaller \dimension" than X: After an exploration of the salient properties of Hausdorff Measure, we proceed to a definition of Hausdorff dimension, a separate idea of size that allows us a more robust comparison between rough subsets of X. Many of the theorems in this report will be summarized in a proof sketch or shown by visual example. For a more rigorous treatment of the same material, we redirect the reader to Gerald B. Folland's Real Analysis: Modern techniques and their applications. Chapter 11 of the 1999 edition served as our primary reference. 2 Hausdorff Measure 2.1 Measuring low-dimensional subsets of X The need for Hausdorff Measure arises from the need to know the size of lower-dimensional subsets of a metric space. This idea is not as exotic as it may sound. In a high school Geometry course, we learn formulas for objects of various dimension embedded in R3: In Figure 1 we see the line segment, the circle, and the sphere, each with it's own idea of size. We call these length, area, and volume, respectively. Figure 1: low-dimensional subsets of R3: 2 4 3 2r πr 3 πr Note that while only volume measures something of the same dimension as the host space, R3; length, and area can still be of interest to us, especially 2 in applications.
    [Show full text]
  • Measure, Integral and Probability
    Marek Capinski´ and Ekkehard Kopp Measure, Integral and Probability Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest To our children; grandchildren: Piotr, Maciej, Jan, Anna; Luk asz Anna, Emily Preface The central concepts in this book are Lebesgue measure and the Lebesgue integral. Their role as standard fare in UK undergraduate mathematics courses is not wholly secure; yet they provide the principal model for the development of the abstract measure spaces which underpin modern probability theory, while the Lebesgue function spaces remain the main source of examples on which to test the methods of functional analysis and its many applications, such as Fourier analysis and the theory of partial differential equations. It follows that not only budding analysts have need of a clear understanding of the construction and properties of measures and integrals, but also that those who wish to contribute seriously to the applications of analytical methods in a wide variety of areas of mathematics, physics, electronics, engineering and, most recently, finance, need to study the underlying theory with some care. We have found remarkably few texts in the current literature which aim explicitly to provide for these needs, at a level accessible to current under- graduates. There are many good books on modern probability theory, and increasingly they recognize the need for a strong grounding in the tools we develop in this book, but all too often the treatment is either too advanced for an undergraduate audience or else somewhat perfunctory. We hope therefore that the current text will not be regarded as one which fills a much-needed gap in the literature! One fundamental decision in developing a treatment of integration is whether to begin with measures or integrals, i.e.
    [Show full text]
  • The Axiom of Choice and Its Implications
    THE AXIOM OF CHOICE AND ITS IMPLICATIONS KEVIN BARNUM Abstract. In this paper we will look at the Axiom of Choice and some of the various implications it has. These implications include a number of equivalent statements, and also some less accepted ideas. The proofs discussed will give us an idea of why the Axiom of Choice is so powerful, but also so controversial. Contents 1. Introduction 1 2. The Axiom of Choice and Its Equivalents 1 2.1. The Axiom of Choice and its Well-known Equivalents 1 2.2. Some Other Less Well-known Equivalents of the Axiom of Choice 3 3. Applications of the Axiom of Choice 5 3.1. Equivalence Between The Axiom of Choice and the Claim that Every Vector Space has a Basis 5 3.2. Some More Applications of the Axiom of Choice 6 4. Controversial Results 10 Acknowledgments 11 References 11 1. Introduction The Axiom of Choice states that for any family of nonempty disjoint sets, there exists a set that consists of exactly one element from each element of the family. It seems strange at first that such an innocuous sounding idea can be so powerful and controversial, but it certainly is both. To understand why, we will start by looking at some statements that are equivalent to the axiom of choice. Many of these equivalences are very useful, and we devote much time to one, namely, that every vector space has a basis. We go on from there to see a few more applications of the Axiom of Choice and its equivalents, and finish by looking at some of the reasons why the Axiom of Choice is so controversial.
    [Show full text]
  • Caratheodory's Extension Theorem
    Caratheodory’s extension theorem DBW August 3, 2016 These notes are meant as introductory notes on Caratheodory’s extension theorem. The presentation is not completely my own work; the presentation heavily relies on the pre- sentation of Noel Vaillant on http://www.probability.net/WEBcaratheodory.pdf. To make the line of arguments as clear as possible, the starting point is the notion of a ring on a topological space and not the notion of a semi-ring. 1 Elementary definitions and properties We fix a topological space Ω. The power set of Ω is denoted P(Ω) and consists of all subsets of Ω. Definition 1. A ring on on Ω is a subset R of P(Ω), such that (i) ∅ ∈ R (ii) A, B ∈ R ⇒ A ∪ B ∈ R (iii) A, B ∈ R ⇒ A \ B ∈ R Definition 2. A σ-algebra on Ω is a subset Σ of P(Ω) such that (i) ∅ ∈ Σ (ii) (An)n∈N ∈ Σ ⇒ ∪n An ∈ Σ (iii) A ∈ Σ ⇒ Ac ∈ Σ Since A∩B = A\(A\B) it follows that any ring on Ω is closed under finite intersections; c c hence any ring is also a semi-ring. Since ∩nAn = (∪nA ) it follows that any σ-algebra is closed under arbitrary intersections. And from A \ B = A ∩ Bc we deduce that any σ-algebra is also a ring. If (Ri)i∈I is a set of rings on Ω then it is clear that ∩I Ri is also a ring on Ω. Let S be any subset of P(Ω), then we call the intersection of all rings on Ω containing S the ring generated by S.
    [Show full text]
  • (Measure Theory for Dummies) UWEE Technical Report Number UWEETR-2006-0008
    A Measure Theory Tutorial (Measure Theory for Dummies) Maya R. Gupta {gupta}@ee.washington.edu Dept of EE, University of Washington Seattle WA, 98195-2500 UWEE Technical Report Number UWEETR-2006-0008 May 2006 Department of Electrical Engineering University of Washington Box 352500 Seattle, Washington 98195-2500 PHN: (206) 543-2150 FAX: (206) 543-3842 URL: http://www.ee.washington.edu A Measure Theory Tutorial (Measure Theory for Dummies) Maya R. Gupta {gupta}@ee.washington.edu Dept of EE, University of Washington Seattle WA, 98195-2500 University of Washington, Dept. of EE, UWEETR-2006-0008 May 2006 Abstract This tutorial is an informal introduction to measure theory for people who are interested in reading papers that use measure theory. The tutorial assumes one has had at least a year of college-level calculus, some graduate level exposure to random processes, and familiarity with terms like “closed” and “open.” The focus is on the terms and ideas relevant to applied probability and information theory. There are no proofs and no exercises. Measure theory is a bit like grammar, many people communicate clearly without worrying about all the details, but the details do exist and for good reasons. There are a number of great texts that do measure theory justice. This is not one of them. Rather this is a hack way to get the basic ideas down so you can read through research papers and follow what’s going on. Hopefully, you’ll get curious and excited enough about the details to check out some of the references for a deeper understanding.
    [Show full text]
  • Measure Theory and Probability
    Measure theory and probability Alexander Grigoryan University of Bielefeld Lecture Notes, October 2007 - February 2008 Contents 1 Construction of measures 3 1.1Introductionandexamples........................... 3 1.2 σ-additive measures ............................... 5 1.3 An example of using probability theory . .................. 7 1.4Extensionofmeasurefromsemi-ringtoaring................ 8 1.5 Extension of measure to a σ-algebra...................... 11 1.5.1 σ-rings and σ-algebras......................... 11 1.5.2 Outermeasure............................. 13 1.5.3 Symmetric difference.......................... 14 1.5.4 Measurable sets . ............................ 16 1.6 σ-finitemeasures................................ 20 1.7Nullsets..................................... 23 1.8 Lebesgue measure in Rn ............................ 25 1.8.1 Productmeasure............................ 25 1.8.2 Construction of measure in Rn. .................... 26 1.9 Probability spaces ................................ 28 1.10 Independence . ................................. 29 2 Integration 38 2.1 Measurable functions.............................. 38 2.2Sequencesofmeasurablefunctions....................... 42 2.3 The Lebesgue integral for finitemeasures................... 47 2.3.1 Simplefunctions............................ 47 2.3.2 Positivemeasurablefunctions..................... 49 2.3.3 Integrablefunctions........................... 52 2.4Integrationoversubsets............................ 56 2.5 The Lebesgue integral for σ-finitemeasure.................
    [Show full text]