DOCSLIB.ORG

Sigma-Algebra from Wikipedia, the Free Encyclopedia Chapter 1

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read 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. Like addition and multiplication, the operations of union and intersection are commutative and associative, and inter- section distributes over unions. However, unlike addition and multiplication, union also distributes over intersection. 2 1.3. THE PRINCIPLE OF DUALITY 3 Two additional pairs of laws involve the special sets called the empty set Ø and the universal set U ; together with the complement operator (AC denotes the complement of A). The empty set has no members, and the universal set has all possible members (in a particular context). Identity laws: • A [ ? = A • A \ U = A Complement laws: • A [ AC = U • A \ AC = ? The identity laws (together with the commutative laws) say that, just like 0 and 1 for addition and multiplication, Ø and U are the identity elements for union and intersection, respectively. Unlike addition and multiplication, union and intersection do not have inverse elements. However the complement laws give the fundamental properties of the somewhat inverse-like unary operation of set complementation. The preceding five pairs of laws—the commutative, associative, distributive, identity and complement laws—encompass all of set algebra, in the sense that every valid proposition in the algebra of sets can be derived from them. Note that if the complement laws are weakened to the rule (AC )C = A , then this is exactly the algebra of proposi- tional linear logic. 1.3 The principle of duality See also: Duality (order theory) Each of the identities stated above is one of a pair of identities such that each can be transformed into the other by interchanging ∪ and ∩, and also Ø and U. These are examples of an extremely important and powerful property of set algebra, namely, the principle of duality for sets, which asserts that for any true statement about sets, the dual statement obtained by interchanging unions and intersections, interchanging U and Ø and reversing inclusions is also true. A statement is said to be self-dual if it is equal to its own dual. 1.4 Some additional laws for unions and intersections The following proposition states six more important laws of set algebra, involving unions and intersections. PROPOSITION 3: For any subsets A and B of a universal set U, the following identities hold: idempotent laws: • A [ A = A • A \ A = A domination laws: • A [ U = U • A \ ? = ? absorption laws: • A [ (A \ B) = A • A \ (A [ B) = A 4 CHAPTER 1. ALGEBRA OF SETS As noted above, each of the laws stated in proposition 3 can be derived from the five fundamental pairs of laws stated above. As an illustration, a proof is given below for the idempotent law for union. Proof: The following proof illustrates that the dual of the above proof is the proof of the dual of the idempotent law for union, namely the idempotent law for intersection. Proof: Intersection can be expressed in terms of set difference : A \ B = A r (A r B) 1.5 Some additional laws for complements The following proposition states five more important laws of set algebra, involving complements. PROPOSITION 4: Let A and B be subsets of a universe U, then: De Morgan’s laws: • (A [ B)C = AC \ BC • (A \ B)C = AC [ BC double complement or Involution law: C • (AC ) = A complement laws for the universal set and the empty set: • ?C = U • U C = ? Notice that the double complement law is self-dual. The next proposition, which is also self-dual, says that the complement of a set is the only set that satisfies the complement laws. In other words, complementation is characterized by the complement laws. PROPOSITION 5: Let A and B be subsets of a universe U, then: uniqueness of complements: • If A [ B = U , and A \ B = ? , then B = AC 1.6 The algebra of inclusion The following proposition says that inclusion, that is the binary relation of one set being a subset of another, is a partial order. PROPOSITION 6: If A, B and C are sets then the following hold: reflexivity: • A ⊆ A antisymmetry: • A ⊆ B and B ⊆ A if and only if A = B transitivity: 1.7. THE ALGEBRA OF RELATIVE COMPLEMENTS 5 • If A ⊆ B and B ⊆ C , then A ⊆ C The following proposition says that for any set S, the power set of S, ordered by inclusion, is a bounded lattice, and hence together with the distributive and complement laws above, show that it is a Boolean algebra. PROPOSITION 7: If A, B and C are subsets of a set S then the following hold: existence of a least element and a greatest element: • ? ⊆ A ⊆ S existence of joins: • A ⊆ A [ B • If A ⊆ C and B ⊆ C , then A [ B ⊆ C existence of meets: • A \ B ⊆ A • If C ⊆ A and C ⊆ B , then C ⊆ A \ B The following proposition says that the statement A ⊆ B is equivalent to various other statements involving unions, intersections and complements. PROPOSITION 8: For any two sets A and B, the following are equivalent: • A ⊆ B • A \ B = A • A [ B = B • A r B = ? • BC ⊆ AC The above proposition shows that the relation of set inclusion can be characterized by either of the operations of set union or set intersection, which means that the notion of set inclusion is axiomatically superfluous. 1.7 The algebra of relative complements The following proposition lists several identities concerning relative complements and set-theoretic differences. PROPOSITION 9: For any universe U and subsets A, B, and C of U, the following identities hold: • C n (A \ B) = (C n A) [ (C n B) • C n (A [ B) = (C n A) \ (C n B) • C n (B n A) = (A \ C) [ (C n B) • (B n A) \ C = (B \ C) n A = B \ (C n A) • (B n A) [ C = (B [ C) n (A n C) • A n A = ? • ? n A = ? • A n ? = A • B n A = AC \ B • (B n A)C = A [ BC • U n A = AC • A n U = ? 6 CHAPTER 1. ALGEBRA OF SETS 1.8 See also • σ-algebra is an algebra of sets, completed to include countably infinite operations. • Axiomatic set theory • Field of sets • Naive set theory • Set (mathematics) 1.9 References • Stoll, Robert R.; Set Theory and Logic, Mineola, N.Y.: Dover Publications (1979) ISBN 0-486-63829-4. “The Algebra of Sets”, pp 16—23 • Courant, Richard, Herbert Robbins, Ian Stewart, What is mathematics?: An Elementary Approach to Ideas and Methods, Oxford University Press US, 1996. ISBN 978-0-19-510519-3. “SUPPLEMENT TO CHAPTER II THE ALGEBRA OF SETS” 1.10 External links • Operations on Sets at ProvenMath Chapter 2 Join (sigma algebra) In mathematics, the join of two sigma algebras over the same set X is the coarsest sigma algebra containing both.[1] 2.1 References [1] Peter Walters, An Introduction to Ergodic Theory (1982) Springer Verlag, ISBN 0-387-90599-5 7 Chapter 3 Measurable function A function is Lebesgue measurable if and only if the preimage of each of the sets [a; 1] is a Lebesgue measurable set. In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration. Specifically, a function between measurable spaces is said to be measurable if the preimage of each measurable set is measurable, analogous to the situation of continuous functions between topological spaces. In probability theory, the sigma algebra often represents the set of available information, and a function (in this context a random variable) is measurable if and only if it represents an outcome that is knowable based on the available information. In contrast, functions that are not Lebesgue measurable are generally considered pathological, at least in the field of analysis. 3.1 Formal definition Let (X, Σ) and (Y, Τ) be measurable spaces, meaning that X and Y are sets equipped with respective sigma algebras Σ and Τ. A function f: X → Y is said to be measurable if the preimage of E under f is in Σ for every E ∈ Τ; i.e.
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]
  • Completeness in Quasi-Pseudometric Spaces—A Survey
    mathematics Article Completeness in Quasi-Pseudometric Spaces—A Survey ¸StefanCobzas Faculty of Mathematics and Computer Science, Babe¸s-BolyaiUniversity, 400 084 Cluj-Napoca, Romania; [email protected] Received:17 June 2020; Accepted: 24 July 2020; Published: 3 August 2020 Abstract: The aim of this paper is to discuss the relations between various notions of sequential completeness and the corresponding notions of completeness by nets or by filters in the setting of quasi-metric spaces. We propose a new definition of right K-Cauchy net in a quasi-metric space for which the corresponding completeness is equivalent to the sequential completeness. In this way we complete some results of R. A. Stoltenberg, Proc. London Math. Soc. 17 (1967), 226–240, and V. Gregori and J. Ferrer, Proc. Lond. Math. Soc., III Ser., 49 (1984), 36. A discussion on nets defined over ordered or pre-ordered directed sets is also included. Keywords: metric space; quasi-metric space; uniform space; quasi-uniform space; Cauchy sequence; Cauchy net; Cauchy filter; completeness MSC: 54E15; 54E25; 54E50; 46S99 1. Introduction It is well known that completeness is an essential tool in the study of metric spaces, particularly for fixed points results in such spaces. The study of completeness in quasi-metric spaces is considerably more involved, due to the lack of symmetry of the distance—there are several notions of completeness all agreing with the usual one in the metric case (see [1] or [2]). Again these notions are essential in proving fixed point results in quasi-metric spaces as it is shown by some papers on this topic as, for instance, [3–6] (see also the book [7]).
    [Show full text]
  • Metric Geometry in a Tame Setting
    University of California Los Angeles Metric Geometry in a Tame Setting A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Erik Walsberg 2015 c Copyright by Erik Walsberg 2015 Abstract of the Dissertation Metric Geometry in a Tame Setting by Erik Walsberg Doctor of Philosophy in Mathematics University of California, Los Angeles, 2015 Professor Matthias J. Aschenbrenner, Chair We prove basic results about the topology and metric geometry of metric spaces which are definable in o-minimal expansions of ordered fields. ii The dissertation of Erik Walsberg is approved. Yiannis N. Moschovakis Chandrashekhar Khare David Kaplan Matthias J. Aschenbrenner, Committee Chair University of California, Los Angeles 2015 iii To Sam. iv Table of Contents 1 Introduction :::::::::::::::::::::::::::::::::::::: 1 2 Conventions :::::::::::::::::::::::::::::::::::::: 5 3 Metric Geometry ::::::::::::::::::::::::::::::::::: 7 3.1 Metric Spaces . 7 3.2 Maps Between Metric Spaces . 8 3.3 Covers and Packing Inequalities . 9 3.3.1 The 5r-covering Lemma . 9 3.3.2 Doubling Metrics . 10 3.4 Hausdorff Measures and Dimension . 11 3.4.1 Hausdorff Measures . 11 3.4.2 Hausdorff Dimension . 13 3.5 Topological Dimension . 15 3.6 Left-Invariant Metrics on Groups . 15 3.7 Reductions, Ultralimits and Limits of Metric Spaces . 16 3.7.1 Reductions of Λ-valued Metric Spaces . 16 3.7.2 Ultralimits . 17 3.7.3 GH-Convergence and GH-Ultralimits . 18 3.7.4 Asymptotic Cones . 19 3.7.5 Tangent Cones . 22 3.7.6 Conical Metric Spaces . 22 3.8 Normed Spaces . 23 4 T-Convexity :::::::::::::::::::::::::::::::::::::: 24 4.1 T-convex Structures .
    [Show full text]
  • 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 ,..
    [Show full text]
  • 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.
    [Show full text]
  • 1 Sets and Classes
    Prerequisites Topological spaces. A set E in a space X is σ-compact if there exists a S1 sequence of compact sets such that E = n=1 Cn. A space X is locally compact if every point of X has a neighborhood whose closure is compact. A subset E of a locally compact space is bounded if there exists a compact set C such that E ⊂ C. Topological groups. The set xE [or Ex] is called a left translation [or right translation.] If Y is a subgroup of X, the sets xY and Y x are called (left and right) cosets of Y . A topological group is a group X which is a Hausdorff space such that the transformation (from X ×X onto X) which sends (x; y) into x−1y is continuous. A class N of open sets containing e in a topological group is a base at e if (a) for every x different form e there exists a set U in N such that x2 = U, (b) for any two sets U and V in N there exists a set W in N such that W ⊂ U \ V , (c) for any set U 2 N there exists a set W 2 N such that V −1V ⊂ U, (d) for any set U 2 N and any element x 2 X, there exists a set V 2 N such that V ⊂ xUx−1, and (e) for any set U 2 N there exists a set V 2 N such that V x ⊂ U. If N is a satisfies the conditions described above, and if the class of all translation of sets of N is taken for a base, then, with respect to the topology so defined, X becomes a topological group.
    [Show full text]
  • Measure Theory and Probability Theory
    Measure Theory and Probability Theory Stéphane Dupraz In this chapter, we aim at building a theory of probabilities that extends to any set the theory of probability we have for finite sets (with which you are assumed to be familiar). For a finite set with N elements Ω = {ω1, ..., ωN }, a probability P takes any n positive numbers p1, ..., pN that sum to one, and attributes to any subset S of Ω the number (S) = P p . Extending this definition to infinitely countable sets such as P i/ωi∈S i N poses no difficulty: we can in the same way assign a positive number to each integer n ∈ N and require that P∞ 1 P n=1 pn = 1. We can then define the probability of a subset S ⊆ N as P(S) = n∈S pn. Things get more complicated when we move to uncountable sets such as the real line R. To be sure, it is possible to assign a positive number to each real number. But how to get from these positive numbers to the probability of any subset of R?2 To get a definition of a probability that applies without a hitch to uncountable sets, we give in the strategy we used for finite and countable sets and start from scratch. The definition of a probability we are going to use was borrowed from measure theory by Kolmogorov in 1933, which explains the title of this chapter. What do probabilities have to do with measurement? Simple: assigning a probability to an event is measuring the likeliness of this event.
    [Show full text]
  • Dynamic Algebras: Examples, Constructions, Applications
    Dynamic Algebras: Examples, Constructions, Applications Vaughan Pratt∗ Dept. of Computer Science, Stanford, CA 94305 Abstract Dynamic algebras combine the classes of Boolean (B ∨ 0 0) and regu- lar (R ∪ ; ∗) algebras into a single finitely axiomatized variety (BR 3) resembling an R-module with “scalar” multiplication 3. The basic result is that ∗ is reflexive transitive closure, contrary to the intuition that this concept should require quantifiers for its definition. Using this result we give several examples of dynamic algebras arising naturally in connection with additive functions, binary relations, state trajectories, languages, and flowcharts. The main result is that free dynamic algebras are residu- ally finite (i.e. factor as a subdirect product of finite dynamic algebras), important because finite separable dynamic algebras are isomorphic to Kripke structures. Applications include a new completeness proof for the Segerberg axiomatization of propositional dynamic logic, and yet another notion of regular algebra. Key words: Dynamic algebra, logic, program verification, regular algebra. This paper originally appeared as Technical Memo #138, Laboratory for Computer Science, MIT, July 1979, and was subsequently revised, shortened, retitled, and published as [Pra80b]. After more than a decade of armtwisting I. N´emeti and H. Andr´eka finally persuaded the author to publish TM#138 itself on the ground that it contained a substantial body of interesting material that for the sake of brevity had been deleted from the STOC-80 version. The principal changes here to TM#138 are the addition of footnotes and the sections at the end on retrospective citations and reflections. ∗This research was supported by the National Science Foundation under NSF grant no.
    [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]
  • The Open Handbook of Formal Epistemology
    THEOPENHANDBOOKOFFORMALEPISTEMOLOGY Richard Pettigrew &Jonathan Weisberg,Eds. THEOPENHANDBOOKOFFORMAL EPISTEMOLOGY Richard Pettigrew &Jonathan Weisberg,Eds. Published open access by PhilPapers, 2019 All entries copyright © their respective authors and licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. LISTOFCONTRIBUTORS R. A. Briggs Stanford University Michael Caie University of Toronto Kenny Easwaran Texas A&M University Konstantin Genin University of Toronto Franz Huber University of Toronto Jason Konek University of Bristol Hanti Lin University of California, Davis Anna Mahtani London School of Economics Johanna Thoma London School of Economics Michael G. Titelbaum University of Wisconsin, Madison Sylvia Wenmackers Katholieke Universiteit Leuven iii For our teachers Overall, and ultimately, mathematical methods are necessary for philosophical progress. — Hannes Leitgeb There is no mathematical substitute for philosophy. — Saul Kripke PREFACE In formal epistemology, we use mathematical methods to explore the questions of epistemology and rational choice. What can we know? What should we believe and how strongly? How should we act based on our beliefs and values? We begin by modelling phenomena like knowledge, belief, and desire using mathematical machinery, just as a biologist might model the fluc- tuations of a pair of competing populations, or a physicist might model the turbulence of a fluid passing through a small aperture. Then, we ex- plore, discover, and justify the laws governing those phenomena, using the precision that mathematical machinery affords. For example, we might represent a person by the strengths of their beliefs, and we might measure these using real numbers, which we call credences. Having done this, we might ask what the norms are that govern that person when we represent them in that way.
    [Show full text]
  • Results Real Analysis I and II, MATH 5453-5463, 2006-2007
    Main results Real Analysis I and II, MATH 5453-5463, 2006-2007 Section Homework Introduction. 1.3 Operations with sets. DeMorgan Laws. 1.4 Proposition 1. Existence of the smallest algebra containing C. 2.5 Open and closed sets. 2.6 Continuous functions. Proposition 18. Hw #1. p.16 #9, 11, 17, 18; p.19 #19. 2.7 Borel sets. p.49 #40, 42, 43; p.53 #53*. 3.2 Outer measure. Proposition 1. Outer measure of an interval. Proposition 2. Subadditivity of the outer measure. Proposition 5. Approximation by open sets. 3.3 Measurable sets. Lemma 6. Measurability of sets of outer measure zero. Lemma 7. Measurability of the union. Hw #2. p.55 #1-4; p.58 # 7, 8. Theorem 10. Measurable sets form a sigma-algebra. Lemma 11. Interval is measurable. Theorem 12. Borel sets are measurable. Proposition 13. Sigma additivity of the measure. Proposition 14. Continuity of the measure. Proposition 15. Approximation by open and closed sets. Hw #3. p.64 #9-11, 13, 14. 3.4 A nonmeasurable set. 3.5 Measurable functions. Proposition 18. Equivalent definitions of measurability. Proposition 19. Sums and products of measurable functions. Theorem 20. Infima and suprema of measurable functions. Hw #4. p.70 #18-22. 3.6 Littlewood's three principles. Egoroff's theorem. Lusin's theorem. 4.2 Prop.2. Lebesgue's integral of a simple function and its props. Lebesgue's integral of a bounded measurable function. Proposition 3. Criterion of integrability. Proposition 5. Properties of integrals of bounded functions. Proposition 6. Bounded convergence theorem. 4.3 Lebesgue integral of a nonnegative function and its properties.
    [Show full text]
  • On Families of Mutually Exclusive Sets
    ANNALS OF MATHEMATICS Vol. 44, No . 2, April, 1943 ON FAMILIES OF MUTUALLY EXCLUSIVE SETS BY P . ERDÖS AND A. TARSKI (Received August 11, 1942) In this paper we shall be concerned with a certain particular problem from the general theory of sets, namely with the problem of the existence of families of mutually exclusive sets with a maximal power . It will turn out-in a rather unexpected way that the solution of these problems essentially involves the notion of the so-called "inaccessible numbers ." In this connection we shall make some general remarks regarding inaccessible numbers in the last section of our paper . §1. FORMULATION OF THE PROBLEM . TERMINOLOGY' The problem in which we are interested can be stated as follows : Is it true that every field F of sets contains a family of mutually exclusive sets with a maximal power, i .e . a family O whose cardinal number is not smaller than the cardinal number of any other family of mutually exclusive sets contained in F . By a field of sets we understand here as usual a family F of sets which to- gether with every two sets X and Y contains also their union X U Y and their difference X - Y (i.e. the set of those elements of X which do not belong to Y) among its elements . A family O is called a family of mutually exclusive sets if no set X of X of O is empty and if any two different sets of O have an empty inter- section. A similar problem can be formulated for other families e .g .
    [Show full text]