DOCSLIB.ORG

1. Σ-Algebras Definition 1. Let X Be Any Set and Let F Be a Collection Of

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
1. Σ-Algebras Definition 1. Let X Be Any Set and Let F Be a Collection Of 1. σ-algebras Definition 1. Let X be any set and let F be a collection of subsets of X. We say that F is a σ-algebra (on X), if it satisfies the following. (1) X 2 F. (2) If A 2 F, then Ac 2 F. 1 (3) If A1;A2; · · · 2 F, a countable collection, then [n=1An 2 F. In the above situation, the elements of F are called measurable sets and X (or more precisely (X; F)) is called a measurable space. Example 1. For any set X, P(X), the set of all subsets of X is a σ-algebra and so is F = f;;Xg. Theorem 1. Let X be any set and let G be any collection of subsets of X. Then there exists a σ-algebra, containing G and smallest with respect to inclusion. Proof. Let S be the collection of all σ-algebras containing G. This set is non-empty, since P(X) 2 S. Then F = \A2SA can easily be checked to have all the properties asserted in the theorem. Remark 1. In the above situation, F is called the σ-algebra generated by G. Definition 2. Let X be a topological space (for example, a metric space) and let B be the σ-algebra generated by the set of all open sets of X. The elements of B are called Borel sets and (X; B), a Borel measurable space. Definition 3. Let (X; F) be a measurable space and let Y be any topo- logical space and let f : X ! Y be any function. We say that f is measurable, if for any open set U ⊂ Y , f −1(U) 2 F. Example 2. (1) If (X; F) is a measurable space with X a topologi- cal space and B ⊂ F, then any continuous function from X to any topological space is measurable. (2) If f : X ! Y is measurable and g : Y ! Z is continuous, g ◦ f is measurable. Theorem 2. Let (X; F) be a measurable space and let f : X ! Y be a function, where Y is any set. (1) If G is defined as the collection of all subsets A of Y such that f −1(A) 2 F, then G is a σ-algebra on Y . (2) If f : X ! [−∞; 1] is any function, f is measurable if and only if f −1((a; 1]) 2 F for any a 2 [−∞; 1]. 1 2 Proof. The first part is mere checking. For the second part, notice that since (a; 1] is open in [−∞; 1], the condition is clearly necessary. To 1 check sufficiency, one notes that [−∞; b) = [n[−∞; b− n ] for any b and [−∞; c] = (c; 1]c. So, one gets that f −1([−∞; b)) 2 F too for any b and thus we get f −1((a; b)) 2 F for any a; b, since (a; b) = [−∞; b) \ (a; 1] and these generate all open sets. Theorem 3. Let fn : X ! [−∞; 1] be a sequence of measurable functions. Then, supfn; inffn; lim supfn; lim inffn are all measurable. −1 −1 Proof. Let g = supfn. Then g ((a; 1]) = [nfn ((a; 1]) and so we are done by the previous theorem. A similar proof applies to infimum. Let gn = supk≥nfk. Then all the gn's are measurable by the previous part. Then lim supfn = infgn and hence it too is measurable. Similar argument applies to lim inf. We get as a corollary the following results. Corollary 1. Let f; g : X ! [−∞; 1] be measurable. Then so are maxff; gg and minff; gg. In particular, writing f + = maxff; 0g and f − = − minff; 0g, we see that these are measurable. Remark 2. In all the above, we have dealt with [−∞; 1] which has the advantage that supremum, lim sup etc. make sense. But, notice that if a; b 2 [−∞; 1], then a + b may not make sense. On the other hand, if we worked with R, then the latter obviously makes sense, but not necessarily the former. In most cases, it should be clear which space we are working in and why. Definition 4. A function f : X ! [0; 1) is called simple, if it takes only finitely many values. Example 3. (1) If A ⊂ X, the characteristic function χA defined as χA(x) = 1 if x 2 A an zero otherwise, is a simple function. (2) More generally, if s is a simple function, let a1; : : : ; an be its −1 Pn finitely many values and let Ai = s (ai). Then s = i=1 aiχAi . Notice that [iAi = X and Ai \ Aj = ; if i 6= j (pairwise disjoint). (3) If s is a simple function as above from a measurable space X, s is measurable if and only if all the Ai's are measurable. P P (4) If s = aiχAi and t = bjχBj are simple (measurable) func- tions, so is s + t. Letting Cij = Ai \ Bj, one can see that P s + t = (ai + bj)χCij . Theorem 4. Let f : X ! [0; 1] be a measurable function. Then there exists simple measurable functions s1 ≤ s2 ≤ · · · ≤ f such that lim sn = f. 3 n −1 i−1 i Proof. For any n and an i such that 1 ≤ i ≤ n2 , let En;i = f ([ 2n ; 2n )) −1 and let Fn = f ([n; 1]). Then these are all measurable sets, pairwise Pn2n i−1 disjoint for a fixed n and cover all of X. So, sn = i=1 2n χEn;i + nχFn is a simple measurable function. Easy to check that sn ≤ sn+1 ≤ f for all n and lim sn = f. 2. Measures Definition 5. Let (X; F) be a measurable space. A (positive) measure on X is a function µ : F! [0; 1] which is countably additive. That is, if A1;A2;::: 2 F is a countable collection of pairwise disjoint sets, 1 P1 then µ([n=1An) = n=1 µ(An). Since µ(A) = 1 for all A 2 F is trivially a measure by the above, to avoid this case, we will always assume that there exists some A 2 F with µ(A) < 1 for our definition. Example 4. (1) Let X be any set and consider the function µ : P(X) ! [0; 1] given by µ(A) is the cardinality of A if A is finite and 1 otherwise. Then µ is a positive measure. This is called the counting measure. (2) Again consider (X; P(X)) and let a 2 X, fixed. Define µ(A) = 1 if a 2 A and zero otherwise. This is a positive measure. Theorem 5. Let µ be a positive measure on (X; F). (1) µ(;) = 0. n (2) If A1;A2;:::;An 2 F, pairwise disjoint, then µ([i=1Ai) = Pn i=1 µ(Ai). (Finite additivity) (3) If A ⊂ B are measurable, µ(A) ≤ µ(B). (Monotonicity) 1 (4) If A1 ⊂ A2 ⊂ · · · are measurable sets and A = [n=1An, then limn!1 µ(An) = µ(A). 1 (5) If A1 ⊃ A2 ⊃ · · · and A = \n=1An with µ(A1) < 1, then lim µ(An) = µ(A). Proof. (1) Let A be such that µ(A) < 1. Let A1 = A and An = ; for n > 1. Then these are pairwise disjoint and its union is A. So, we get µ(A) = µ(A) + P µ(;). Rest is clear, since we can cancel µ(A) < 1 from bot sides. (2) This is clear by taking Am = ; for m > n and using the first. (3) We can write B = A [ (B − A), pairwise disjoint and thus µ(B) = µ(A) + µ(B − A) ≥ µ(A). (4) Let Bn = An − An−1 for n > 1 and B1 = A1. Then fBng are 1 n pairwise disjoint and [n=1Bn = A. Also, [i=1Bi = An. So, we Pn P1 get µ(An) = i=1 µ(Bi) and µ(A) = n=1 µ(Bn). 4 1 (5) For this, let Cn = A1 − An. Then Cn ⊂ Cn+1 and [n=1Cn = A1 −A. So, from the previous part, we have lim µ(Cn) = µ(A1 − A) = µ(A1) − µ(A), since A ⊂ A1. We also have by the same reason, µ(Cn) = µ(A1) − µ(An). So, taking limits, we get, lim µ(Cn) = µ(A1) − lim(An) = µ(A1) − µ(A) and canceling µ(A1) < 1, we are done. 3. Lebesgue Integral Definition 6. Let (X; F; µ) be a measure space and let s : X ! [0; 1) P be a simple positive measurable function. Writing s = aiχA , define R P i X sdµ = aiµ(Ai), where we use the convention, 0 · µ(A) = 0 even if µ(A) = 1. If 0 ≤ f is any positive measurable function, we define R R X fdµ to be the supremum of X sdµ for simple measurable functions 0 ≤ s ≤ f. Remark 3. The above gives two possible definitions for the integral of a simple measurable function, but it is clear that the definitions coincide. We collect a few easy facts which follow from the definitions. Theorem 6. Let (X; F; µ) be a measure space. All functions men- tioned below are measurable and so are all the subsets of X. R R (1) If 0 ≤ f ≤ g, then X fdµ ≤ X gdµ. (2) If A ⊂ B in F, for 0 ≤ f on X, we have, R fdµ ≤ R fdµ. R RA B (3) If 0 ≤ f and 0 ≤ c < 1, then X cfdµ = c X fdµ. (4) If f(x) = 0 for all x 2 A, R fdµ = 0, even if µ(A) = 1. A R (5) If µ(A) = 0, for any measurable positive f, A fdµ = 0 even if f(x) = 1 for some or all x 2 A.
Load more

Recommended publications
  • 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.
    [Show full text]
  • Distributions:The Evolutionof a Mathematicaltheory
    Historia Mathematics 10 (1983) 149-183 DISTRIBUTIONS:THE EVOLUTIONOF A MATHEMATICALTHEORY BY JOHN SYNOWIEC INDIANA UNIVERSITY NORTHWEST, GARY, INDIANA 46408 SUMMARIES The theory of distributions, or generalized func- tions, evolved from various concepts of generalized solutions of partial differential equations and gener- alized differentiation. Some of the principal steps in this evolution are described in this paper. La thgorie des distributions, ou des fonctions g&&alis~es, s'est d&eloppeg 2 partir de divers con- cepts de solutions g&&alis6es d'gquations aux d&i- &es partielles et de diffgrentiation g&&alis6e. Quelques-unes des principales &apes de cette &olution sont d&rites dans cet article. Die Theorie der Distributionen oder verallgemein- erten Funktionen entwickelte sich aus verschiedenen Begriffen von verallgemeinerten Lasungen der partiellen Differentialgleichungen und der verallgemeinerten Ableitungen. In diesem Artikel werden einige der wesentlichen Schritte dieser Entwicklung beschrieben. 1. INTRODUCTION The founder of the theory of distributions is rightly con- sidered to be Laurent Schwartz. Not only did he write the first systematic paper on the subject [Schwartz 19451, which, along with another paper [1947-19481, already contained most of the basic ideas, but he also wrote expository papers on distributions for electrical engineers [19481. Furthermore, he lectured effec- tively on his work, for example, at the Canadian Mathematical Congress [1949]. He showed how to apply distributions to partial differential equations [19SObl and wrote the first general trea- tise on the subject [19SOa, 19511. Recognition of his contri- butions came in 1950 when he was awarded the Fields' Medal at the International Congress of Mathematicians, and in his accep- tance address to the congress Schwartz took the opportunity to announce his celebrated "kernel theorem" [195Oc].
    [Show full text]
  • A Convenient Category for Higher-Order Probability Theory
    A Convenient Category for Higher-Order Probability Theory Chris Heunen Ohad Kammar Sam Staton Hongseok Yang University of Edinburgh, UK University of Oxford, UK University of Oxford, UK University of Oxford, UK Abstract—Higher-order probabilistic programming languages 1 (defquery Bayesian-linear-regression allow programmers to write sophisticated models in machine 2 let let sample normal learning and statistics in a succinct and structured way, but step ( [f ( [s ( ( 0.0 3.0)) 3 sample normal outside the standard measure-theoretic formalization of proba- b ( ( 0.0 3.0))] 4 fn + * bility theory. Programs may use both higher-order functions and ( [x] ( ( s x) b)))] continuous distributions, or even define a probability distribution 5 (observe (normal (f 1.0) 0.5) 2.5) on functions. But standard probability theory does not handle 6 (observe (normal (f 2.0) 0.5) 3.8) higher-order functions well: the category of measurable spaces 7 (observe (normal (f 3.0) 0.5) 4.5) is not cartesian closed. 8 (observe (normal (f 4.0) 0.5) 6.2) Here we introduce quasi-Borel spaces. We show that these 9 (observe (normal (f 5.0) 0.5) 8.0) spaces: form a new formalization of probability theory replacing 10 (predict :f f))) measurable spaces; form a cartesian closed category and so support higher-order functions; form a well-pointed category and so support good proof principles for equational reasoning; and support continuous probability distributions. We demonstrate the use of quasi-Borel spaces for higher-order functions and proba- bility by: showing that a well-known construction of probability theory involving random functions gains a cleaner expression; and generalizing de Finetti’s theorem, that is a crucial theorem in probability theory, to quasi-Borel spaces.
    [Show full text]
  • Jointly Measurable and Progressively Measurable Stochastic Processes
    Jointly measurable and progressively measurable stochastic processes Jordan Bell [email protected] Department of Mathematics, University of Toronto June 18, 2015 1 Jointly measurable stochastic processes d Let E = R with Borel E , let I = R≥0, which is a topological space with the subspace topology inherited from R, and let (Ω; F ;P ) be a probability space. For a stochastic process (Xt)t2I with state space E, we say that X is jointly measurable if the map (t; !) 7! Xt(!) is measurable BI ⊗ F ! E . For ! 2 Ω, the path t 7! Xt(!) is called left-continuous if for each t 2 I, Xs(!) ! Xt(!); s " t: We prove that if the paths of a stochastic process are left-continuous then the stochastic process is jointly measurable.1 Theorem 1. If X is a stochastic process with state space E and all the paths of X are left-continuous, then X is jointly measurable. Proof. For n ≥ 1, t 2 I, and ! 2 Ω, let 1 n X Xt (!) = 1[k2−n;(k+1)2−n)(t)Xk2−n (!): k=0 n Each X is measurable BI ⊗ F ! E : for B 2 E , 1 n [ −n −n f(t; !) 2 I × Ω: Xt (!) 2 Bg = [k2 ; (k + 1)2 ) × fXk2−n 2 Bg: k=0 −n −n Let t 2 I. For each n there is a unique kn for which t 2 [kn2 ; (kn +1)2 ), and n −n −n thus Xt (!) = Xkn2 (!). Furthermore, kn2 " t, and because s 7! Xs(!) is n −n left-continuous, Xkn2 (!) ! Xt(!).
    [Show full text]
  • Arxiv:1404.7630V2 [Math.AT] 5 Nov 2014 Asoffsae,Se[S4 P8,Vr6.Isedof Instead Ver66]
    PROPER BASE CHANGE FOR SEPARATED LOCALLY PROPER MAPS OLAF M. SCHNÜRER AND WOLFGANG SOERGEL Abstract. We introduce and study the notion of a locally proper map between topological spaces. We show that fundamental con- structions of sheaf theory, more precisely proper base change, pro- jection formula, and Verdier duality, can be extended from contin- uous maps between locally compact Hausdorff spaces to separated locally proper maps between arbitrary topological spaces. Contents 1. Introduction 1 2. Locally proper maps 3 3. Proper direct image 5 4. Proper base change 7 5. Derived proper direct image 9 6. Projection formula 11 7. Verdier duality 12 8. The case of unbounded derived categories 15 9. Remindersfromtopology 17 10. Reminders from sheaf theory 19 11. Representability and adjoints 21 12. Remarks on derived functors 22 References 23 arXiv:1404.7630v2 [math.AT] 5 Nov 2014 1. Introduction The proper direct image functor f! and its right derived functor Rf! are defined for any continuous map f : Y → X of locally compact Hausdorff spaces, see [KS94, Spa88, Ver66]. Instead of f! and Rf! we will use the notation f(!) and f! for these functors. They are embedded into a whole collection of formulas known as the six-functor-formalism of Grothendieck. Under the assumption that the proper direct image functor f(!) has finite cohomological dimension, Verdier proved that its ! derived functor f! admits a right adjoint f . Olaf Schnürer was supported by the SPP 1388 and the SFB/TR 45 of the DFG. Wolfgang Soergel was supported by the SPP 1388 of the DFG. 1 2 OLAFM.SCHNÜRERANDWOLFGANGSOERGEL In this article we introduce in 2.3 the notion of a locally proper map between topological spaces and show that the above results even hold for arbitrary topological spaces if all maps whose proper direct image functors are involved are locally proper and separated.
    [Show full text]
  • A Class of Symmetric Difference-Closed Sets Related to Commuting Involutions John Campbell
    A class of symmetric difference-closed sets related to commuting involutions John Campbell To cite this version: John Campbell. A class of symmetric difference-closed sets related to commuting involutions. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2017, Vol 19 no. 1. hal-01345066v4 HAL Id: hal-01345066 https://hal.archives-ouvertes.fr/hal-01345066v4 Submitted on 18 Mar 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Discrete Mathematics and Theoretical Computer Science DMTCS vol. 19:1, 2017, #8 A class of symmetric difference-closed sets related to commuting involutions John M. Campbell York University, Canada received 19th July 2016, revised 15th Dec. 2016, 1st Feb. 2017, accepted 10th Feb. 2017. Recent research on the combinatorics of finite sets has explored the structure of symmetric difference-closed sets, and recent research in combinatorial group theory has concerned the enumeration of commuting involutions in Sn and An. In this article, we consider an interesting combination of these two subjects, by introducing classes of symmetric difference-closed sets of elements which correspond in a natural way to commuting involutions in Sn and An. We consider the natural combinatorial problem of enumerating symmetric difference-closed sets consisting of subsets of sets consisting of pairwise disjoint 2-subsets of [n], and the problem of enumerating symmetric difference-closed sets consisting of elements which correspond to commuting involutions in An.
    [Show full text]
  • 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.
    [Show full text]
  • Shape Analysis, Lebesgue Integration and Absolute Continuity Connections
    NISTIR 8217 Shape Analysis, Lebesgue Integration and Absolute Continuity Connections Javier Bernal This publication is available free of charge from: https://doi.org/10.6028/NIST.IR.8217 NISTIR 8217 Shape Analysis, Lebesgue Integration and Absolute Continuity Connections Javier Bernal Applied and Computational Mathematics Division Information Technology Laboratory This publication is available free of charge from: https://doi.org/10.6028/NIST.IR.8217 July 2018 INCLUDES UPDATES AS OF 07-18-2018; SEE APPENDIX U.S. Department of Commerce Wilbur L. Ross, Jr., Secretary National Institute of Standards and Technology Walter Copan, NIST Director and Undersecretary of Commerce for Standards and Technology ______________________________________________________________________________________________________ This Shape Analysis, Lebesgue Integration and publication Absolute Continuity Connections Javier Bernal is National Institute of Standards and Technology, available Gaithersburg, MD 20899, USA free of Abstract charge As shape analysis of the form presented in Srivastava and Klassen’s textbook “Functional and Shape Data Analysis” is intricately related to Lebesgue integration and absolute continuity, it is advantageous from: to have a good grasp of the latter two notions. Accordingly, in these notes we review basic concepts and results about Lebesgue integration https://doi.org/10.6028/NIST.IR.8217 and absolute continuity. In particular, we review fundamental results connecting them to each other and to the kind of shape analysis, or more generally, functional data analysis presented in the aforeme- tioned textbook, in the process shedding light on important aspects of all three notions. Many well-known results, especially most results about Lebesgue integration and some results about absolute conti- nuity, are presented without proofs.
    [Show full text]
  • Extremal Dependence Concepts
    Extremal dependence concepts Giovanni Puccetti1 and Ruodu Wang2 1Department of Economics, Management and Quantitative Methods, University of Milano, 20122 Milano, Italy 2Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON N2L3G1, Canada Journal version published in Statistical Science, 2015, Vol. 30, No. 4, 485{517 Minor corrections made in May and June 2020 Abstract The probabilistic characterization of the relationship between two or more random variables calls for a notion of dependence. Dependence modeling leads to mathematical and statistical challenges and recent developments in extremal dependence concepts have drawn a lot of attention to probability and its applications in several disciplines. The aim of this paper is to review various concepts of extremal positive and negative dependence, including several recently established results, reconstruct their history, link them to probabilistic optimization problems, and provide a list of open questions in this area. While the concept of extremal positive dependence is agreed upon for random vectors of arbitrary dimensions, various notions of extremal negative dependence arise when more than two random variables are involved. We review existing popular concepts of extremal negative dependence given in literature and introduce a novel notion, which in a general sense includes the existing ones as particular cases. Even if much of the literature on dependence is focused on positive dependence, we show that negative dependence plays an equally important role in the solution of many optimization problems. While the most popular tool used nowadays to model dependence is that of a copula function, in this paper we use the equivalent concept of a set of rearrangements.
    [Show full text]
  • Natural Numerosities of Sets of Tuples
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 367, Number 1, January 2015, Pages 275–292 S 0002-9947(2014)06136-9 Article electronically published on July 2, 2014 NATURAL NUMEROSITIES OF SETS OF TUPLES MARCO FORTI AND GIUSEPPE MORANA ROCCASALVO Abstract. We consider a notion of “numerosity” for sets of tuples of natural numbers that satisfies the five common notions of Euclid’s Elements, so it can agree with cardinality only for finite sets. By suitably axiomatizing such a no- tion, we show that, contrasting to cardinal arithmetic, the natural “Cantorian” definitions of order relation and arithmetical operations provide a very good algebraic structure. In fact, numerosities can be taken as the non-negative part of a discretely ordered ring, namely the quotient of a formal power series ring modulo a suitable (“gauge”) ideal. In particular, special numerosities, called “natural”, can be identified with the semiring of hypernatural numbers of appropriate ultrapowers of N. Introduction In this paper we consider a notion of “equinumerosity” on sets of tuples of natural numbers, i.e. an equivalence relation, finer than equicardinality, that satisfies the celebrated five Euclidean common notions about magnitudines ([8]), including the principle that “the whole is greater than the part”. This notion preserves the basic properties of equipotency for finite sets. In particular, the natural Cantorian definitions endow the corresponding numerosities with a structure of a discretely ordered semiring, where 0 is the size of the emptyset, 1 is the size of every singleton, and greater numerosity corresponds to supersets. This idea of “Euclidean” numerosity has been recently investigated by V.
    [Show full text]
  • LEBESGUE MEASURE and L2 SPACE. Contents 1. Measure Spaces 1 2. Lebesgue Integration 2 3. L2 Space 4 Acknowledgments 9 References
    LEBESGUE MEASURE AND L2 SPACE. ANNIE WANG Abstract. This paper begins with an introduction to measure spaces and the Lebesgue theory of measure and integration. Several important theorems regarding the Lebesgue integral are then developed. Finally, we prove the completeness of the L2(µ) space and show that it is a metric space, and a Hilbert space. Contents 1. Measure Spaces 1 2. Lebesgue Integration 2 3. L2 Space 4 Acknowledgments 9 References 9 1. Measure Spaces Definition 1.1. Suppose X is a set. Then X is said to be a measure space if there exists a σ-ring M (that is, M is a nonempty family of subsets of X closed under countable unions and under complements)of subsets of X and a non-negative countably additive set function µ (called a measure) defined on M . If X 2 M, then X is said to be a measurable space. For example, let X = Rp, M the collection of Lebesgue-measurable subsets of Rp, and µ the Lebesgue measure. Another measure space can be found by taking X to be the set of all positive integers, M the collection of all subsets of X, and µ(E) the number of elements of E. We will be interested only in a special case of the measure, the Lebesgue measure. The Lebesgue measure allows us to extend the notions of length and volume to more complicated sets. Definition 1.2. Let Rp be a p-dimensional Euclidean space . We denote an interval p of R by the set of points x = (x1; :::; xp) such that (1.3) ai ≤ xi ≤ bi (i = 1; : : : ; p) Definition 1.4.
    [Show full text]
  • MTH 304: General Topology Semester 2, 2017-2018
    MTH 304: General Topology Semester 2, 2017-2018 Dr. Prahlad Vaidyanathan Contents I. Continuous Functions3 1. First Definitions................................3 2. Open Sets...................................4 3. Continuity by Open Sets...........................6 II. Topological Spaces8 1. Definition and Examples...........................8 2. Metric Spaces................................. 11 3. Basis for a topology.............................. 16 4. The Product Topology on X × Y ...................... 18 Q 5. The Product Topology on Xα ....................... 20 6. Closed Sets.................................. 22 7. Continuous Functions............................. 27 8. The Quotient Topology............................ 30 III.Properties of Topological Spaces 36 1. The Hausdorff property............................ 36 2. Connectedness................................. 37 3. Path Connectedness............................. 41 4. Local Connectedness............................. 44 5. Compactness................................. 46 6. Compact Subsets of Rn ............................ 50 7. Continuous Functions on Compact Sets................... 52 8. Compactness in Metric Spaces........................ 56 9. Local Compactness.............................. 59 IV.Separation Axioms 62 1. Regular Spaces................................ 62 2. Normal Spaces................................ 64 3. Tietze's extension Theorem......................... 67 4. Urysohn Metrization Theorem........................ 71 5. Imbedding of Manifolds..........................
    [Show full text]