Appendix a Basics of Measure Theory
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Measure-Theoretic Probability I
Measure-Theoretic Probability I Steven P.Lalley Winter 2017 1 1 Measure Theory 1.1 Why Measure Theory? There are two different views – not necessarily exclusive – on what “probability” means: the subjectivist view and the frequentist view. To the subjectivist, probability is a system of laws that should govern a rational person’s behavior in situations where a bet must be placed (not necessarily just in a casino, but in situations where a decision must be made about how to proceed when only imperfect information about the outcome of the decision is available, for instance, should I allow Dr. Scissorhands to replace my arthritic knee by a plastic joint?). To the frequentist, the laws of probability describe the long- run relative frequencies of different events in “experiments” that can be repeated under roughly identical conditions, for instance, rolling a pair of dice. For the frequentist inter- pretation, it is imperative that probability spaces be large enough to allow a description of an experiment, like dice-rolling, that is repeated infinitely many times, and that the mathematical laws should permit easy handling of limits, so that one can make sense of things like “the probability that the long-run fraction of dice rolls where the two dice sum to 7 is 1/6”. But even for the subjectivist, the laws of probability should allow for description of situations where there might be a continuum of possible outcomes, or pos- sible actions to be taken. Once one is reconciled to the need for such flexibility, it soon becomes apparent that measure theory (the theory of countably additive, as opposed to merely finitely additive measures) is the only way to go. -
2.1 the Algebra of Sets
Chapter 2 I Abstract Algebra 83 part of abstract algebra, sets are fundamental to all areas of mathematics and we need to establish a precise language for sets. We also explore operations on sets and relations between sets, developing an “algebra of sets” that strongly resembles aspects of the algebra of sentential logic. In addition, as we discussed in chapter 1, a fundamental goal in mathematics is crafting articulate, thorough, convincing, and insightful arguments for the truth of mathematical statements. We continue the development of theorem-proving and proof-writing skills in the context of basic set theory. After exploring the algebra of sets, we study two number systems denoted Zn and U(n) that are closely related to the integers. Our approach is based on a widely used strategy of mathematicians: we work with specific examples and look for general patterns. This study leads to the definition of modified addition and multiplication operations on certain finite subsets of the integers. We isolate key axioms, or properties, that are satisfied by these and many other number systems and then examine number systems that share the “group” properties of the integers. Finally, we consider an application of this mathematics to check digit schemes, which have become increasingly important for the success of business and telecommunications in our technologically based society. Through the study of these topics, we engage in a thorough introduction to abstract algebra from the perspective of the mathematician— working with specific examples to identify key abstract properties common to diverse and interesting mathematical systems. 2.1 The Algebra of Sets Intuitively, a set is a “collection” of objects known as “elements.” But in the early 1900’s, a radical transformation occurred in mathematicians’ understanding of sets when the British philosopher Bertrand Russell identified a fundamental paradox inherent in this intuitive notion of a set (this paradox is discussed in exercises 66–70 at the end of this section). -
Integral Representation of a Linear Functional on Function Spaces
INTEGRAL REPRESENTATION OF LINEAR FUNCTIONALS ON FUNCTION SPACES MEHDI GHASEMI Abstract. Let A be a vector space of real valued functions on a non-empty set X and L : A −! R a linear functional. Given a σ-algebra A, of subsets of X, we present a necessary condition for L to be representable as an integral with respect to a measure µ on X such that elements of A are µ-measurable. This general result then is applied to the case where X carries a topological structure and A is a family of continuous functions and naturally A is the Borel structure of X. As an application, short solutions for the full and truncated K-moment problem are presented. An analogue of Riesz{Markov{Kakutani representation theorem is given where Cc(X) is replaced with whole C(X). Then we consider the case where A only consists of bounded functions and hence is equipped with sup-norm. 1. Introduction A positive linear functional on a function space A ⊆ RX is a linear map L : A −! R which assigns a non-negative real number to every function f 2 A that is globally non-negative over X. The celebrated Riesz{Markov{Kakutani repre- sentation theorem states that every positive functional on the space of continuous compactly supported functions over a locally compact Hausdorff space X, admits an integral representation with respect to a regular Borel measure on X. In symbols R L(f) = X f dµ, for all f 2 Cc(X). Riesz's original result [12] was proved in 1909, for the unit interval [0; 1]. -
Complex Measures 1 11
Tutorial 11: Complex Measures 1 11. Complex Measures In the following, (Ω, F) denotes an arbitrary measurable space. Definition 90 Let (an)n≥1 be a sequence of complex numbers. We a say that ( n)n≥1 has the permutation property if and only if, for ∗ ∗ +∞ 1 all bijections σ : N → N ,theseries k=1 aσ(k) converges in C Exercise 1. Let (an)n≥1 be a sequence of complex numbers. 1. Show that if (an)n≥1 has the permutation property, then the same is true of (Re(an))n≥1 and (Im(an))n≥1. +∞ 2. Suppose an ∈ R for all n ≥ 1. Show that if k=1 ak converges: +∞ +∞ +∞ + − |ak| =+∞⇒ ak = ak =+∞ k=1 k=1 k=1 1which excludes ±∞ as limit. www.probability.net Tutorial 11: Complex Measures 2 Exercise 2. Let (an)n≥1 be a sequence in R, such that the series +∞ +∞ k=1 ak converges, and k=1 |ak| =+∞.LetA>0. We define: + − N = {k ≥ 1:ak ≥ 0} ,N = {k ≥ 1:ak < 0} 1. Show that N + and N − are infinite. 2. Let φ+ : N∗ → N + and φ− : N∗ → N − be two bijections. Show the existence of k1 ≥ 1 such that: k1 aφ+(k) ≥ A k=1 3. Show the existence of an increasing sequence (kp)p≥1 such that: kp aφ+(k) ≥ A k=kp−1+1 www.probability.net Tutorial 11: Complex Measures 3 for all p ≥ 1, where k0 =0. 4. Consider the permutation σ : N∗ → N∗ defined informally by: φ− ,φ+ ,...,φ+ k ,φ− ,φ+ k ,...,φ+ k ,.. -
Appendix A. Measure and Integration
Appendix A. Measure and integration We suppose the reader is familiar with the basic facts concerning set theory and integration as they are presented in the introductory course of analysis. In this appendix, we review them briefly, and add some more which we shall need in the text. Basic references for proofs and a detailed exposition are, e.g., [[ H a l 1 ]] , [[ J a r 1 , 2 ]] , [[ K F 1 , 2 ]] , [[ L i L ]] , [[ R u 1 ]] , or any other textbook on analysis you might prefer. A.1 Sets, mappings, relations A set is a collection of objects called elements. The symbol card X denotes the cardi- nality of the set X. The subset M consisting of the elements of X which satisfy the conditions P1(x),...,Pn(x) is usually written as M = { x ∈ X : P1(x),...,Pn(x) }.A set whose elements are certain sets is called a system or family of these sets; the family of all subsystems of a given X is denoted as 2X . The operations of union, intersection, and set difference are introduced in the standard way; the first two of these are commutative, associative, and mutually distributive. In a { } system Mα of any cardinality, the de Morgan relations , X \ Mα = (X \ Mα)and X \ Mα = (X \ Mα), α α α α are valid. Another elementary property is the following: for any family {Mn} ,whichis { } at most countable, there is a disjoint family Nn of the same cardinality such that ⊂ \ ∪ \ Nn Mn and n Nn = n Mn.Theset(M N) (N M) is called the symmetric difference of the sets M,N and denoted as M #N. -
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. -
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(!). -
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. -
Version of 21.8.15 Chapter 43 Topologies and Measures II The
Version of 21.8.15 Chapter 43 Topologies and measures II The first chapter of this volume was ‘general’ theory of topological measure spaces; I attempted to distinguish the most important properties a topological measure can have – inner regularity, τ-additivity – and describe their interactions at an abstract level. I now turn to rather more specialized investigations, looking for features which offer explanations of the behaviour of the most important spaces, radiating outwards from Lebesgue measure. In effect, this chapter consists of three distinguishable parts and two appendices. The first three sections are based on ideas from descriptive set theory, in particular Souslin’s operation (§431); the properties of this operation are the foundation for the theory of two classes of topological space of particular importance in measure theory, the K-analytic spaces (§432) and the analytic spaces (§433). The second part of the chapter, §§434-435, collects miscellaneous results on Borel and Baire measures, looking at the ways in which topological properties of a space determine properties of the measures it carries. In §436 I present the most important theorems on the representation of linear functionals by integrals; if you like, this is the inverse operation to the construction of integrals from measures in §122. The ideas continue into §437, where I discuss spaces of signed measures representing the duals of spaces of continuous functions, and topologies on spaces of measures. The first appendix, §438, looks at a special topic: the way in which the patterns in §§434-435 are affected if we assume that our spaces are not unreasonably complex in a rather special sense defined in terms of measures on discrete spaces. -
On Stochastic Distributions and Currents
NISSUNA UMANA INVESTIGAZIONE SI PUO DIMANDARE VERA SCIENZIA S’ESSA NON PASSA PER LE MATEMATICHE DIMOSTRAZIONI LEONARDO DA VINCI vol. 4 no. 3-4 2016 Mathematics and Mechanics of Complex Systems VINCENZO CAPASSO AND FRANCO FLANDOLI ON STOCHASTIC DISTRIBUTIONS AND CURRENTS msp MATHEMATICS AND MECHANICS OF COMPLEX SYSTEMS Vol. 4, No. 3-4, 2016 dx.doi.org/10.2140/memocs.2016.4.373 ∩ MM ON STOCHASTIC DISTRIBUTIONS AND CURRENTS VINCENZO CAPASSO AND FRANCO FLANDOLI Dedicated to Lucio Russo, on the occasion of his 70th birthday In many applications, it is of great importance to handle random closed sets of different (even though integer) Hausdorff dimensions, including local infor- mation about initial conditions and growth parameters. Following a standard approach in geometric measure theory, such sets may be described in terms of suitable measures. For a random closed set of lower dimension with respect to the environment space, the relevant measures induced by its realizations are sin- gular with respect to the Lebesgue measure, and so their usual Radon–Nikodym derivatives are zero almost everywhere. In this paper, how to cope with these difficulties has been suggested by introducing random generalized densities (dis- tributions) á la Dirac–Schwarz, for both the deterministic case and the stochastic case. For the last one, mean generalized densities are analyzed, and they have been related to densities of the expected values of the relevant measures. Ac- tually, distributions are a subclass of the larger class of currents; in the usual Euclidean space of dimension d, currents of any order k 2 f0; 1;:::; dg or k- currents may be introduced. -
[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. -
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.