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Proofs in Elementary Analysis by Branko´Curgus January 15, 2016
Proofs in Elementary Analysis by Branko Curgus´ January 15, 2016 Contents Chapter 1. Introductory Material 5 1.1. Goals 5 1.2. Strategies 5 1.3. Mathematics and logic 6 1.4. Proofs 8 1.5. Four basic ingredients of a Proof 10 Chapter 2. Sets and functions 11 2.1. Sets 11 2.2. Functions 14 2.3. Cardinality of sets 17 Chapter 3. The Set R of real numbers 19 3.1. Axioms of a field 19 3.2. Axioms of order in a field 21 3.3. Three functions: the unit step, the sign and the absolute value 23 3.4. Intervals 25 3.5. Bounded sets. Minimum and Maximum 28 3.6. The set N 29 3.7. Examples and Exercises related to N 32 3.8. Finite sets, infinite sets, countable sets 34 3.9. More on countable sets 37 3.10. The sets Z and Q 41 3.11. The Completeness axiom 42 3.12. More on the sets N, Z and Q 45 3.13. Infimum and supremum 47 3.14. The topology of R 49 3.15. The topology of R2 51 Chapter 4. Sequences in R 53 4.1. Definitions and examples 53 4.2. Bounded sequences 54 4.3. Thedefinitionofaconvergentsequence 55 4.4. Finding N(ǫ)foraconvergentsequence 57 4.5. Two standard sequences 60 4.6. Non-convergent sequences 61 4.7. Convergence and boundedness 61 4.8. Algebra of limits of convergent sequences 61 4.9. Convergent sequences and the order in R 62 4.10. Themonotonicconvergencetheorem 63 3 4 CONTENTS 4.11. -
A Set Optimization Approach to Utility Maximization Under Transaction Costs
A set optimization approach to utility maximization under transaction costs Andreas H. Hamel,∗ Sophie Qingzhen Wangy Abstract A set optimization approach to multi-utility maximization is presented, and duality re- sults are obtained for discrete market models with proportional transaction costs. The novel approach admits to obtain results for non-complete preferences, where the formulas derived closely resemble but generalize the scalar case. Keywords. utility maximization, non-complete preference, multi-utility representation, set optimization, duality theory, transaction costs JEL classification. C61, G11 1 Introduction In this note, we propose a set-valued approach to utility maximization for market models with transaction costs. For finite probability spaces and a one-period set-up, we derive results which resemble very closely the scalar case as discussed in [4, Theorem 3.2.1]. This is far beyond other approaches in which only scalar utility functions are used as, for example, in [1, 3], where a complete preference for multivariate position is assumed. As far as we are aware of, there is no argument justifying such strong assumption, and it does not seem appropriate for market models with transaction costs. On the other hand, recent results on multi-utility representations as given, among others, in [5] lead to the question how to formulate and solve an \expected multi-utility" maximiza- tion problem. The following optimistic goal formulated by Bosi and Herden in [2] does not seem achievable since, in particular, there is no satisfactory multi-objective duality which matches the power of the scalar version: `Moreover, as it reduces finding the maximal ele- ments in a given subset of X with respect to to a multi-objective optimization problem (cf. -
Integration 1 Measurable Functions
Integration References: Bass (Real Analysis for Graduate Students), Folland (Real Analysis), Athreya and Lahiri (Measure Theory and Probability Theory). 1 Measurable Functions Let (Ω1; F1) and (Ω2; F2) be measurable spaces. Definition 1 A function T :Ω1 ! Ω2 is (F1; F2)-measurable if for every −1 E 2 F2, T (E) 2 F1. Terminology: If (Ω; F) is a measurable space and f is a real-valued func- tion on Ω, it's called F-measurable or simply measurable, if it is (F; B(<))- measurable. A function f : < ! < is called Borel measurable if the σ-algebra used on the domain and codomain is B(<). If the σ-algebra on the domain is Lebesgue, f is called Lebesgue measurable. Example 1 Measurability of a function is related to the σ-algebras that are chosen in the domain and codomain. Let Ω = f0; 1g. If the σ-algebra is P(Ω), every real valued function is measurable. Indeed, let f :Ω ! <, and E 2 B(<). It is clear that f −1(E) 2 P(Ω) (this includes the case where f −1(E) = ;). However, if F = f;; Ωg is the σ-algebra, only the constant functions are measurable. Indeed, if f(x) = a; 8x 2 Ω, then for any Borel set E containing a, f −1(E) = Ω 2 F. But if f is a function s.t. f(0) 6= f(1), then, any Borel set E containing f(0) but not f(1) will satisfy f −1(E) = f0g 2= F. 1 It is hard to check for measurability of a function using the definition, because it requires checking the preimages of all sets in F2. -
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. -
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. -
Supremum and Infimum of Set of Real Numbers
ISSN (Online) :2319-8753 ISSN (Print) : 2347-6710 International Journal of Innovative Research in Science, Engineering and Technology (An ISO 3297: 2007 Certified Organization) Vol. 4, Issue 9, September 2015 Supremum and Infimum of Set of Real Numbers M. Somasundari1, K. Janaki2 1,2Department of Mathematics, Bharath University, Chennai, Tamil Nadu, India ABSTRACT: In this paper, we study supremum and infimum of set of real numbers. KEYWORDS: Upper bound, Lower bound, Largest member, Smallest number, Least upper bound, Greatest Lower bound. I. INTRODUCTION Let S denote a set(a collection of objects). The notation x∈ S means that the object x is in the set S, and we write x ∉ S to indicate that x is not in S.. We assume there exists a nonempty set R of objects, called real numbers, which satisfy the ten axioms listed below. The axioms fall in anatural way into three groups which we refer to as the field axioms, the order axioms, and the completeness axiom( also called the least-upper-bound axiom or the axiom of continuity)[10]. Along with the R of real numbers we assume the existence of two operations, called addition and multiplication, such that for every pair of real numbers x and ythe sum x+y and the product xy are real numbers uniquely dtermined by x and y satisfying the following axioms[9]. ( In the axioms that appear below, x, y, z represent arbitrary real numbers unless something is said to the contrary.). In this section, we define axioms[6]. The field axioms Axiom 1. x+y = y+x, xy =yx (commutative laws) Axiom 2. -
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. -
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. -
Old Notes from Warwick, Part 1
Measure Theory, MA 359 Handout 1 Valeriy Slastikov Autumn, 2005 1 Measure theory 1.1 General construction of Lebesgue measure In this section we will do the general construction of σ-additive complete measure by extending initial σ-additive measure on a semi-ring to a measure on σ-algebra generated by this semi-ring and then completing this measure by adding to the σ-algebra all the null sets. This section provides you with the essentials of the construction and make some parallels with the construction on the plane. Throughout these section we will deal with some collection of sets whose elements are subsets of some fixed abstract set X. It is not necessary to assume any topology on X but for simplicity you may imagine X = Rn. We start with some important definitions: Definition 1.1 A nonempty collection of sets S is a semi-ring if 1. Empty set ? 2 S; 2. If A 2 S; B 2 S then A \ B 2 S; n 3. If A 2 S; A ⊃ A1 2 S then A = [k=1Ak, where Ak 2 S for all 1 ≤ k ≤ n and Ak are disjoint sets. If the set X 2 S then S is called semi-algebra, the set X is called a unit of the collection of sets S. Example 1.1 The collection S of intervals [a; b) for all a; b 2 R form a semi-ring since 1. empty set ? = [a; a) 2 S; 2. if A 2 S and B 2 S then A = [a; b) and B = [c; d). -
Some Topologies Connected with Lebesgue Measure Séminaire De Probabilités (Strasbourg), Tome 5 (1971), P
SÉMINAIRE DE PROBABILITÉS (STRASBOURG) JOHN B. WALSH Some topologies connected with Lebesgue measure Séminaire de probabilités (Strasbourg), tome 5 (1971), p. 290-310 <http://www.numdam.org/item?id=SPS_1971__5__290_0> © Springer-Verlag, Berlin Heidelberg New York, 1971, tous droits réservés. L’accès aux archives du séminaire de probabilités (Strasbourg) (http://portail. mathdoc.fr/SemProba/) implique l’accord avec les conditions générales d’utili- sation (http://www.numdam.org/conditions). Toute utilisation commerciale ou im- pression systématique est constitutive d’une infraction pénale. Toute copie ou im- pression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ SOME TOPOLOGIES CONNECTED WITH LEBESGUE MEASURE J. B. Walsh One of the nettles flourishing in the nether regions of the field of continuous parameter processes is the quantity lim sup X. s-~ t When one most wants to use it, he can’t show it is measurable. A theorem of Doob asserts that one can always modify the process slightly so that it becomes in which case lim sup is the same separable, X~ s-t as its less relative lim where D is a countable prickly sup X, set. If, as sometimes happens, one is not free to change the process, the usual procedure is to use the countable limit anyway and hope that the process is continuous. Chung and Walsh [3] used the idea of an essential limit-that is a limit ignoring sets of Lebesgue measure and found that it enjoyed the pleasant measurability and separability properties of the countable limit in addition to being translation invariant. -
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................. -
Measurable Functions and Simple Functions
Measure theory class notes - 8 September 2010, class 9 1 Measurable functions and simple functions The class of all real measurable functions on (Ω, A ) is too vast to study directly. We identify ways to study them via simpler functions or collections of functions. Recall that L is the set of all measurable functions from (Ω, A ) to R and E⊆L are all the simple functions. ∞ Theorem. Suppose f ∈ L is bounded. Then there exists an increasing sequence {fn}n=1 in E whose uniform limit is f. Proof. First assume that f takes values in [0, 1). We divide [0, 1) into 2n intervals and use this to construct fn: n− 2 1 k − k k+1 fn = 1f 1 n , n 2n ([ 2 2 )) Xk=0 k k+1 k Whenever f takes a value in 2n , 2n , fn takes the value 2n . We have • For all n, fn ≤ f. 1 • For all n, x, |fn(x) − f(x)|≤ 2n . This is clear from the construction. • fn ∈E, since fn is a finite linear combination of indicator functions of sets in A . k 2k 2k+1 • fn ≤ fn+1: For any x, if fn(x)= 2n , then fn+1(x) ∈ 2n+1 , 2n+1 . ∞ So {fn}n=1 is an increasing sequence in E converging uniformly to f. Now for a general f, if the image of f lies in [a,b), then let f − a1 g = Ω b − a (note that a1Ω is the constant function a) ∞ Im g ⊆ [0, 1), so by the above we have {gn}n=1 from E increasing uniformly to g.