DOCSLIB.ORG

Chapter 0 Review of Set Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Chapter 0 Review of Set Theory Chapter 0 Review of set theory Set theory plays a central role in the theory of probability. Thus, we will open this course with a quick review of those notions of set theory which will be used repeatedly. 0.1 Sets Throughout this chapter, let X be a set, which we will call the whole space. The elements of X will often be called points. If A is a subset of X and x is a point in X, then the notation x A means that x belongs to A, i.e., it is one of the points in A. If x does not belong to ∈ A, we write x A. Whenever the identity of the whole space X is clear, we will refer to subsets of X ∈ simples as sets. If A and B are sets, the notations A B or B A mean that A is a subset of B, i.e.,⊂ every point in ⊃A is also a point in B. Two sets are called equal if they contain exactly the same set of points, i.e., A B and B A. ⊂ ⊂ 6 Chapter 0 When considering the collection of all subsets of X, we always include the empty set, which is denoted by . By definition, x X, x . For every set A, it always holds that∀ ∈ ∈ A X. We will often consider sets of sets. To⊂ avoid⊂ confusion, we will use the term a collection (42&!) of sets, rather then a set of sets; however remember, a collection is nothing but a set. Thus, is is a collection of sets and A is a set, then A means that the set A is one of the elements of the collection of sets . The notation A , on the other hand, is meaningless.C ∈ C C Example⊂ C : Let X be a whole space. The collection of all its subsets is denoted by 2X; it is called the power set of X (%8'(% ;7&"8). This notation is suggestive, as if X has finite cardinality X , then 2X 2 X . = Definition 0.1 A sequence An of sets is called increasing (%-&3) if ▲▲▲ ( )A1 A2 A3 . It is called decreasing (;$9&*) if ⊂ ⊂ ⊂ A1 A2 A3 . In either case it is called monotone.⊃ ⊃ ⊃ To denote sets, we will use curly brackets. For example, if x, y X, then x, y ∈ n denotes the set whose only elements are{ x}and y. If xk k 1 is a finite sequence of distinct points in X, we will denote the set whose only elements are the elements = of that sequence by ( ) xk k 1,...,n . { ∶ = } Review of set theory 7 If xk is an infinite sequence of distinct points in X, we will denote the set whose only elements are the elements of that sequence by ( ) xk k N . { ∶ ∈ } Suppose that for every x X, ⇡ x is a proposition assuming values either True or False. Then, ∈ ( ) x X ⇡ x denotes the set of all points in X{for∈ which∶ ( the)} proposition ⇡ x is True. More generally, if ⇡1 x ,⇡2 x ,... is a sequence of propositions regarding the point x, then ( ) ( ) ( ) x X ⇡1 x ,⇡2 x ,... denotes the set of all points{ in ∈X for∶ which( ) all( the) propositions} ⇡k x are True. ( ) Example: Let X N and for n N let = ∈ True n is even ⇡ n False n is odd. ( ) = Then, n N ⇡ n denotes the set of even integers. In{ many∈ ∶ instances,( )} we will use the more intuitive notation n N n is even . { ∈ ∶ } As a tautological, yet useful remark: every set A can be expressed using the curly ▲▲▲ bracket notation as A x X x A . Finally, for every x X, x is a= subset{ ∈ of∶X, i.e.,∈ } an element of 2X; such a subset is called a singleton (0&$*(*). Be careful to distinguish x from x . ∈ { } { } 8 Chapter 0 0.2 Unions and intersections Let be a collection of subsets of X. The subset of X including all those points that belong to at least one set of the collection is called the union ($&(*!) of the collection,C and it is denoted by C or A A . In the case where the collection C includes{ only∶ two∈ C} sets, e.g., A, B , we simply write C A B. C = { } If C = ∪ Ai i 1,...,n is a finite collection of sets, weC = write{ ∶ = } n Ai. i 1 C = If = Ai i N is a countable collection of sets,C we= write{ ∶ ∈ } Ai. ∞ i 1 C = Finally, if = Aγ γ Γ Γ is a non-countable collection ofC sets,= { where∶ ∈is} an index set, we write Aγ. γ Γ C = ∈ By convention if is an empty collection of sets, then C . It is easy to see that whenever A B, thenC = ⊂ A B B. ∪ = Review of set theory 9 In particular, it always holds that A A and A X X. Moreover, if An is an∪ increasing= sequence of sets,∪ then= ( ) n Ak An. k 1 = = Let be a collection of subsets of X. The subset of X including all those points that belong to all sets of the collection is called the intersection (+&;*() of the collection,C and it is denoted by C or A A . In the case where the collection C includes{ only∶ two∈ C} sets, e.g., A, B , we simply write C A B. C = { } If C = ∩ Ai i 1,...,n is a finite collection of sets, weC = write{ ∶ = } n Ai. i 1 C = = If Ai i N is a countable collection of sets,C we= write{ ∶ ∈ } Ai. ∞ i 1 C = = Finally, if Aγ γ Γ is a non-countable collection ofC sets,= { where∶ Γ∈is} an index set, we write Aγ. γ Γ C = ∈ 10 Chapter 0 By convention if is an empty collection of sets then C X. C = It is easy to see that whenever A B, then ⊂ A B A. In particular, it always holds that ∩ = A and A X A. Moreover, if An is a decreasing∩ = sequence of sets,∩ then= ( ) n Ak An. k 1 = = Definition 0.2 Two sets A and B are called disjoint (.*9') if they have no points in common, i.e., A B .Adisjoint collection of sets is a collection of sets such that every two distinct sets in this collection are disjoint. ∩ = Since unions of disjoint collections of sets play an important role in the theory of probability, we will denote the union of a disjoint collection by . For example, A B denotes the union of the disjoint sets A and B. Unions and intersections satisfy the following important properties: 1. Both are commutative and associative. 2. Intersections are distributive over unions, A B C A B A C . 3. Unions are distributive∩ over( ∪ intersections,) = ( ∩ ) ∪ ( ∩ ) A B C A B A C . ∪ ( ∩ ) = ( ∪ ) ∩ ( ∪ ) Review of set theory 11 0.3 Complements Let A be a set. It complement (.*-:/) is the set of points which do not belong to A, Ac x x A . Complementation satisfies the following= { ∶ properties:∈ } 1. For all sets A, A Ac X and A Ac . c c 2. For all sets A, A ∪A. = ∩ = 3. c X and Xc . ( ) = 4. If A B then Ac Bc. = = 5. For every (possibly non-countable) collection of sets, ⊂ ⊃ c A A Ac C A , and ({ ∶ ∈ C}) = { ∶ ∈ C} A A c Ac A . Let A and B be sets. The( di↵{erence∶ between∈ C}) =Aand{ B∶is the∈ C set} of all those points that are in A but not in B, namely, A B x x A and x B A Bc. = { ∶ ∈ ∈ } = ∩ 0.4 Limits Let An be a sequence of sets. The superior limit (0&*-3 -&"#) of this sequence is defined as the set of points that belong to infinitely many of those sets: ( ) lim sup An x for all k there exists an n k such that x An . n = { ∶ ≥ ∈ } If x lim supn An we say that x An infinitely often (%1:1 05&!"). The inferior limit (0&;(; -&"#) of this sequence is defined as the set of points that ∈ ∈ belong to all but finitely many of those sets: lim inf An x there exists k such that x An for all n k . n = { ∶ ∈ ≥ } 12 Chapter 0 If x lim supn An we say that x An eventually ($*/; 9"$ -: &5&2"). Note that both superior and inferior limits always exist. Also, it always holds that ∈ ∈ lim inf An lim sup An. n n ⊂ In the event that the superior limit and the inferior limit of a sequence of sets coincide, we call this set the limit of the sequence, namely, lim An lim sup An lim inf An. n n n = = By definition, for k N, ∈ An x there exists an n k such that x An , ∞ n k = { ∶ ≥ ∈ } hence, = An lim sup An. ∞ ∞ k 1 n k n Likewise, for k N, = = = ∈ An x x An for all n k , ∞ n k = { ∶ ∈ ≥ } hence, = An lim inf An. ∞ ∞ n k 1 n k = = = Proposition 0.1 If An is a monotone sequence of sets, then it has a limit. Specif- ically, if An is increasing then ( ) ( ) lim An An. n ∞ n 1 = = and if An is decreasing then ( ) lim An An. n ∞ n 1 = = Review of set theory 13 Proof : Let An be increasing. Then for every k N, An An and Ak. ( ) ∞ ∞ ∈ ∞ n k n 1 n k Thus, = = = = lim sup An An An, ∞ ∞ ∞ n k 1 n 1 n 1 and = = = = = lim inf An An. n ∞ k 1 Let An be decreasing. Then for every k =N=, An An and Ak. ( ) ∞ ∞ ∈ ∞ n k n 1 n k Thus, = = = = lim sup An Ak, ∞ n k 1 and = = lim inf An An An.
Load more

Recommended publications
  • 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).
    [Show full text]
  • An Elementary Approach to Boolean Algebra
    Eastern Illinois University The Keep Plan B Papers Student Theses & Publications 6-1-1961 An Elementary Approach to Boolean Algebra Ruth Queary Follow this and additional works at: https://thekeep.eiu.edu/plan_b Recommended Citation Queary, Ruth, "An Elementary Approach to Boolean Algebra" (1961). Plan B Papers. 142. https://thekeep.eiu.edu/plan_b/142 This Dissertation/Thesis is brought to you for free and open access by the Student Theses & Publications at The Keep. It has been accepted for inclusion in Plan B Papers by an authorized administrator of The Keep. For more information, please contact [email protected]. r AN ELEr.:ENTARY APPRCACH TC BCCLF.AN ALGEBRA RUTH QUEAHY L _J AN ELE1~1ENTARY APPRCACH TC BC CLEAN ALGEBRA Submitted to the I<:athematics Department of EASTERN ILLINCIS UNIVERSITY as partial fulfillment for the degree of !•:ASTER CF SCIENCE IN EJUCATION. Date :---"'f~~-----/_,_ffo--..i.-/ _ RUTH QUEARY JUNE 1961 PURPOSE AND PLAN The purpose of this paper is to provide an elementary approach to Boolean algebra. It is designed to give an idea of what is meant by a Boclean algebra and to supply the necessary background material. The only prerequisite for this unit is one year of high school algebra and an open mind so that new concepts will be considered reason­ able even though they nay conflict with preconceived ideas. A mathematical science when put in final form consists of a set of undefined terms and unproved propositions called postulates, in terrrs of which all other concepts are defined, and from which all other propositions are proved.
    [Show full text]
  • Equivalents to the Axiom of Choice and Their Uses A
    EQUIVALENTS TO THE AXIOM OF CHOICE AND THEIR USES A Thesis Presented to The Faculty of the Department of Mathematics California State University, Los Angeles In Partial Fulfillment of the Requirements for the Degree Master of Science in Mathematics By James Szufu Yang c 2015 James Szufu Yang ALL RIGHTS RESERVED ii The thesis of James Szufu Yang is approved. Mike Krebs, Ph.D. Kristin Webster, Ph.D. Michael Hoffman, Ph.D., Committee Chair Grant Fraser, Ph.D., Department Chair California State University, Los Angeles June 2015 iii ABSTRACT Equivalents to the Axiom of Choice and Their Uses By James Szufu Yang In set theory, the Axiom of Choice (AC) was formulated in 1904 by Ernst Zermelo. It is an addition to the older Zermelo-Fraenkel (ZF) set theory. We call it Zermelo-Fraenkel set theory with the Axiom of Choice and abbreviate it as ZFC. This paper starts with an introduction to the foundations of ZFC set the- ory, which includes the Zermelo-Fraenkel axioms, partially ordered sets (posets), the Cartesian product, the Axiom of Choice, and their related proofs. It then intro- duces several equivalent forms of the Axiom of Choice and proves that they are all equivalent. In the end, equivalents to the Axiom of Choice are used to prove a few fundamental theorems in set theory, linear analysis, and abstract algebra. This paper is concluded by a brief review of the work in it, followed by a few points of interest for further study in mathematics and/or set theory. iv ACKNOWLEDGMENTS Between the two department requirements to complete a master's degree in mathematics − the comprehensive exams and a thesis, I really wanted to experience doing a research and writing a serious academic paper.
    [Show full text]
  • Sets, Functions, Relations
    Mathematics for Computer Scientists Lecture notes for the module G51MCS Venanzio Capretta University of Nottingham School of Computer Science Chapter 4 Sets, Functions and Relations 4.1 Sets Sets are such a basic notion in mathematics that the only way to ‘define’ them is by synonyms like collection, class, grouping and so on. A set is completely characterised by the elements it contains. There are two main ways of defining a set: (1) by explicitly listing all its elements or (2) by giving a property that all elements must satisfy. Here is a couple of examples, the first is a set of fruits given by listing its elements, the second is a set of natural numbers given by specifying a property: A = {apple, banana, cherry, peach} E = {n ∈ N | 2 divides n} The definition of E can be read: E is the set of those natural numbers that are divisible by 2, that is, it’s the set of all even naturals. The first definition method can be used only if the set has a finite number of elements. In some cases both methods can be used to define the same set, as in this example: 2 {n ∈ N | n is odd ∧ n + n ≤ 100} = {1, 3, 5, 7, 9} To say that a certain object x is an element of a set S, we write x ∈ S. To say that it isn’t an element of S we write x 6∈ S. For example: apple ∈ A 7 6∈ E strawberry 6∈ A 8 ∈ E If every element of a set X is also an element of another set Y , then we say that X is a subset of Y and we write this symbolically as X ⊆ Y .
    [Show full text]
  • Set Theory Background for Probability Defining Sets (A Very Naïve Approach)
    Set theory background for probability Defining sets (a very naïve approach) A set is a collection of distinct objects. The objects within a set may be arbitrary, with the order of objects within them having no significance. Here are several examples, demonstrating the above properties: 2,3,5,7,11,13,17,19 (the set of all prime numbers smaller than 20) , , , , , ,′′ The brackets … surround the objects that belong to a certain set. Objects are separated by commas. Note that: 1,2,3 1,3,2 (there is no ordering of objects within a set) 1,2,3,3 1,2,3 (sets contain one copy of each distinct object) Working with sets 1. Membership If an object is a member of a set we write: ∈. If an object is not a member of a set we write: ∉. 2. Subsets Assume that we have a set and another set , such that every element in is also a member of . Then we can write: ⊆. We say that is a subset of . Formally, we can write: ∀∈∈ (any that is a member of is a member of ) No need to worry: we will usually not encounter this formal logic notation. Notice that the sets of , may actually be equal. If they are not, there must be some member of that is not a member of . In this case we can write: ⊂ i.e. is a true subset of , or in logic notation: ∀ ∈ ∈ ∧∃∈∈ (any that is in is in AND there exists a in that is not in ). 3. Set equality Sets , are equal if and only if ⊆ and ⊆.
    [Show full text]
  • Measures in Boolean Algebras
    MEASURES IN BOOLEAN ALGEBRAS BY ALFRED HORN AND ALFRED TARSKI Introduction. By a measure on a Boolean algebra we understand as usual a function defined over all elements of the algebra, assuming finite non-nega- tive real numbers as values (but not identically vanishing), and satisfying the condition of finite additivity: the function value at a sum of two disjoint elements equals the sum of the function values at these elements. By includ- ing in this definition some additional conditions we arrive at special kinds of measures: two-valued measures, strictly positive measures, and countably addi- tive measures. In §1 we concern ourselves with the general notion of measure; we show how a measure—and in particular a two-valued measure—defined on a sub- algebra of a given Boolean algebra, or even on an arbitrary subset of the algebra, can be extended to the whole algebra. In §§2 and 3 we consider spe- cial kinds of measures, in fact, strictly positive and countably additive meas- ures, and we establish several partial criteria (necessary or sufficient condi- tions) for their existence. Some of the results stated in this paper can be ob- tained in a roundabout way from what is to be found in the literature, but they will be provided here with rather simple and direct proofs, without applying notions not involved in the formulation of the results. Terminology and symbolism^). Given any two sets A and B, we denote their union (or sum) by A\JB, their intersection (or product) by AC\B, their difference by A —B.
    [Show full text]
  • Field ( -Algebra)
    σ-field (σ-algebra) We will shown later that the class 2X of all subsets can be too large for our quantities of interest to hold, with consistency, across all member of the class. In this section, we present smaller classes which have several “nice” properties. • Let X be a (nonempty) set (universal), and F is a family/set of subsets of X. F is called a field (or algebra ) (of sets) provided that 1) ∅∈F (or X ∈F or F is non-empty.) 2) AA∈⇒FFc ∈. 3) AB, ∈⇒∪∈FF A B . • F ⊂ 2X 3* ) AB, ∈⇒∩∈FF A B c Proof. AB∩=()cc A ∪ B • AB,\∈⇒FF A B ∈ Proof. A \ BA=∩c B. • Property (1) can be replaced by X ∈F because they are equivalent under property (2). • Property (1) can be replaced by (*) F is non-empty. This is because, (2) and (3) implies that (1) is equivalent to (*). Proof. “⇒” Trivial. ∅∈F ⇒ F is non-empty. “⇐” If F is non-empty, then there exists A∈ F. Now, by (2) , we have Ac is also in F. Hence, by (3), XAA=∪c ∈F . By (2), ∅ =∈X c F . • Property (3) can be replaced by (3* ) : AB, ∈ FF⇒∩∈ A B . Proof. Under (2), we will show that (3) is equivalent to (3* ) . Suppose AB, ∈ F ; by (2), we have ABcc, ∈ F . Now, suppose we have (3), then c ABcc∪∈F . Apply (2) again, we have AB∩ =∪( Acc B) ∈F . Similarly, suppose we have (3* ) . Then, ABcc∩ ∈ F . Apply (2) and we c then have AB∪=( Acc ∩ B) ∈F . • AB,\∈⇒Δ=FF A B() A B ∪ B\ A ∈.
    [Show full text]
  • Sigma-Algebra from Wikipedia, the Free Encyclopedia Chapter 1
    Sigma-algebra From Wikipedia, the free encyclopedia Chapter 1 Algebra of sets The algebra of sets defines the properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evalu- ating expressions, and performing calculations, involving these operations and relations. Any set of sets closed under the set-theoretic operations forms a Boolean algebra with the join operator being union, the meet operator being intersection, and the complement operator being set complement. 1.1 Fundamentals The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation “less than or equal” is reflexive, antisymmetric and transitive, so is the set relation of “subset”. It is the algebra of the set-theoretic operations of union, intersection and complementation, and the relations of equality and inclusion. For a basic introduction to sets see the article on sets, for a fuller account see naive set theory, and for a full rigorous axiomatic treatment see axiomatic set theory. 1.2 The fundamental laws of set algebra The binary operations of set union ( [ ) and intersection ( \ ) satisfy many identities. Several of these identities or “laws” have well established names. Commutative laws: • A [ B = B [ A • A \ B = B \ A Associative laws: • (A [ B) [ C = A [ (B [ C) • (A \ B) \ C = A \ (B \ C) Distributive laws: • A [ (B \ C) = (A [ B) \ (A [ C) • A \ (B [ C) = (A \ B) [ (A \ C) The analogy between unions and intersections of sets, and addition and multiplication of numbers, is quite striking.
    [Show full text]
  • Sets and Probability
    CHAPTER 1 Sets and Probability In a survey of 200 people that had just returned from a trip to Europe, the following informa- tion was gathered. • 142 visited England • 95 visited Italy • 65 visited Germany • 70 visited both England and Italy • 50 visited both England and Germany • 30 visited both Italy and Germany • 20 visited all three of these countries How many went to England but not Italy or Germany? We will learn how to solve puzzles like this in the second section of the chapter when count- ing the elements in a set is discussed. Exerpt from Applied Finite Mathematics by Tomastik and Epstein (c) 2008 Cengage Learning 1.1 Introduction to Sets 13 1.1 Introduction to Sets This section discusses operations on sets and the laws governing these set opera- tions. These are fundamental notions that will be used throughout the remainder of this text. In the next two chapters we will see that probability and statistics are based on counting the elements in sets and manipulating set operations. Thus we first need to understand clearly the notion of sets and their operations. ✧ The Language of Sets We begin here with some definitions of the language and notation used when working with sets. The most basic definition is “What is a set?” A set is a HISTORICAL NOTE collection of items. These items are referred to as the elements or members of George Boole, 1815–1864 the set. For example, the set containing the numbers 1, 2, and 3 would be written {1,2,3}. Notice that the set is contained in curly brackets.
    [Show full text]
  • An Introduction to Boolean Algebras
    California State University, San Bernardino CSUSB ScholarWorks Electronic Theses, Projects, and Dissertations Office of aduateGr Studies 12-2016 AN INTRODUCTION TO BOOLEAN ALGEBRAS Amy Schardijn California State University - San Bernardino Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd Part of the Algebra Commons Recommended Citation Schardijn, Amy, "AN INTRODUCTION TO BOOLEAN ALGEBRAS" (2016). Electronic Theses, Projects, and Dissertations. 421. https://scholarworks.lib.csusb.edu/etd/421 This Thesis is brought to you for free and open access by the Office of aduateGr Studies at CSUSB ScholarWorks. It has been accepted for inclusion in Electronic Theses, Projects, and Dissertations by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. An Introduction to Boolean Algebras A Thesis Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degree Master of Arts in Mathematics by Amy Michiel Schardijn December 2016 An Introduction to Boolean Algebras A Thesis Presented to the Faculty of California State University, San Bernardino by Amy Michiel Schardijn December 2016 Approved by: Dr. Giovanna Llosent, Committee Chair Date Dr. Jeremy Aikin, Committee Member Dr. Corey Dunn, Committee Member Dr. Charles Stanton, Chair, Dr. Corey Dunn Department of Mathematics Graduate Coordinator, Department of Mathematics iii Abstract This thesis discusses the topic of Boolean algebras. In order to build intuitive understanding of the topic, research began with the investigation of Boolean algebras in the area of Abstract Algebra. The content of this initial research used a particular nota- tion. The ideas of partially ordered sets, lattices, least upper bounds, and greatest lower bounds were used to define the structure of a Boolean algebra.
    [Show full text]
  • Section 2.3. the Σ-Algebra of Lebesgue Measurable Sets
    2.3. Lebesgue Measurable Sets 1 Section 2.3. The σ-Algebra of Lebesgue Measurable Sets Note. In Theorem 2.18 we will see that there are disjoint sets A and B such that m∗(A ∪· B) < m∗(A) + m∗(B) and so m∗ is not countably additive (it is not even finitely additive). We must have countable additivity in our measure in order to use it for integration. To accomplish this, we restrict m∗ to a smaller class than P(R) on which m∗ is countably additive. As we will see, the following condition will yield a collection of sets on which m∗ is countably additive. Definition. Any set E is (Lebesgue) measurable if for all A ⊂ R, m∗(A) = m∗(A ∩ E) + m∗(A ∩ Ec). This is called the Carath´eodory splitting condition. Note. It certainly isn’t clear that the Carath´eodory splitting condition leads to countable additivity, but notice that it does involve the sum of measures of two disjoint sets (A ∩ E and A ∩ Ec). Note. The Carath´eodory splitting condition, in a sense, requires us to “check” all ℵ2 subsets A of R in order to establish the measurability of a set E. It is shown in Problem 2.20 that if m∗(E) < ∞, then E is measurable if and only if m∗((a, b)) = m∗((a, b)∩E)+m∗((a, b)∩Ec). This is a significant result, since there are “only” ℵ1 intervals of the form (a, b). 2.3. Lebesgue Measurable Sets 2 Note. Recall that we define a function to be Riemann integrable if the upper Riemann integral equals the lower Riemann integral.
    [Show full text]
  • A MINIMAL MODEL for SET THEORY in the Proof of The
    A MINIMAL MODEL FOR SET THEORY BY PAUL J. COHEN1 Communicated by Paul R. Halmos, March 22, 1963 In the proof of the consistency of the Continuum Hypothesis and the Axiom of Choice with the other axioms of set theory, Gödel [l] introduced the notion of a constructible set and showed that the con­ structible sets form a model for set theory. These sets are intuitively those which can be reached by means of a transfinite sequence of several simple operations. He then showed that the Axiom of Choice and Continuum Hypothesis held in the collection of constructible sets. If the original universe of sets is sufficiently rich in ordinal numbers, it will follow that every set is constructible, in which case we say that the Axiom of Constructibility is satisfied. This axiom implies the two axioms previously mentioned. However, from one point of view it may seem that this notion of constructibility does not intuitively cor­ respond to what is meant by constructive since it may happen that all sets in the universe are constructive. In this paper we show that a more restricted notion of "construction" will yield a class of sets which form a minimal model for set theory. In this manner we prove the consistency of a stronger form of the Axiom of Constructibility. We observe that the idea of a minimal collection of objects satisfying certain axioms is well known in mathematics, for example, in group theory one often considers the subgroup generated by a collection of elements, and in measure theory we define the Borel sets as the small­ est <r-algebra of sets containing the open sets.
    [Show full text]