DOCSLIB.ORG

(V, ·) Be a Banach Space. Show That There

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
(V, ·) Be a Banach Space. Show That There Math541, Spring 2013, Homework #6 Solution Due to Friday, April 12, 2013 # 1. Let (V; k·k) be a Banach space. Show that there exists a compact topological space X and there exists an isometric inclusion ι :(V; k · k) ! (C(X); k · k1): This shows that every Banach space can be identified with a norm closed subspace of some C(X) space. Proof: Let us first recall that we can define the weak* topology on the dual space ∗ − V , and by Banach-Alaoglu theorem that the closed unit ball X = BV ∗ (0; 1) of V ∗ is compact with respect to the weak* topology on V ∗. Let us consider this compact topological space (X; τw∗ ). Next, we consider the canonical isometric inclusion ^ : x 2 V ! x^ 2 V ∗∗, where we definex ^(f) = f(x) for all f 2 V ∗. According to the definition of weak* topology on V ∗, for every x 2 V ,x ^ is a weak* continuous linear functional on V ∗, and thus the restriction ofx ^ to X, which is still denoted byx ^, is a continuous function on X (with respect to the weak* topology). This defines a map ι : x 2 V ! x^ 2 (C(X); k · k1): It is clear that this is a well-defined linear map, and it is an isometry injection by the Hahn-Banach extension theorem. # 2. Find all extreme points of the closed unit ball of C([0; 1]) = ff : [0; 1] ! R; continuous functionsg: Proof: Let K be the closed unit ball of C([0; 1]). It is easy to see that the constant function 1 and −1 are extreme points of K. Indeed, if there exist continuous functions f; g : [0; 1] ! [−1; 1] in K and 0 < α < 1 such that 1 = αf(x) + (1 − α)g(x) for all x 2 [0; 1], we must have f = 1 = g. Now let f be an extreme point of K. We must have kfk1 = supfjf(x)j : x 2 [0; 1]g = 1 and thus there exists a point c 2 [0; 1] such that jf(c)j = 1. If f(c) = 1, we claim that f(x) = 1 for all x 2 [0; 1]. Suppose not. We can find a point x0 2 (0; 1) such that 0 < f(x0) < 1. Let " = 1 − f(x0) > 0. By the continuity, there exists a sufficiently small δ > 0 such that (x0 − δ; x0 + δ) ⊆ (0; 1) and " " 1 − < f(x) < 1 + 2 2 for all x 2 (x0 − δ; x0 + δ). Now it is known from Urysohn's lemma, there exists a continuous function h : [0; 1] ! [0; 1] such that h(x0) = 1 and h(x) = 0 for all " x2 = (x0 − δ; x0 + δ). In this case, we can get two distinct functions f1 = f + 2 h " 1 1 and f2 = f − 2 h in K such that f = 2 f1 + 2 f2. This contradicts to f to be an extreme point of K. Therefore, if f(c) = 1, we must have f = 1. Similarly, we can prove that if f(c) = −1 for some c 2 [0; 1], we must have f = −1. These are the only extreme points of K. 1 2 # 3. Show that 1. there exists no extreme point in the closed unit ball of L1([0; 1]), 2. L1([0; 1]) is not a dual Banach space. Proof of Part 1: Let K be the closed unit ball of L1([0; 1]) and let f be an arbitrary R function in K with kfk1 = [0;1] jf(x)jdx = 1. By the (absolute) continuity of R R 1 F (t) = [0;t] jf(x)jdx, we can find c 2 [0; 1] such that F (c) = [0;c] jf(x)jdx = 2 . We can consider the distinct integrable functions g(x) = 2f(x)χ[0;c] and h(x) = 1 2f(x)χ[c;1]. It is easy to see that kgk1 = khk1 = 1 and f = 2 (g + h). This shows that any function in the closed unit ball K can not be an extreme point. Therefore, K has no extreme point at all. Proof of Part 2: Since the closed unit ball of a dual Banach space is compact with respect to the weak* topology, it must contain some extreme points by the Krein-Milman theorem. Since the closed unit ball of L1([0; 1]) has no extreme point, L1([0; 1]) can not be a dual Banach space. # 4. Let (X; τ) be a compact topological space. It is known from the Riesz representation theorem that we have the isometric isomorphism M(X) = C(X)∗, where M(X) denotes the sapce of all real valued Radon measures on X. Therefore, we obtain the weak* topology on M(X) under the representation Z µ(f) = f(x)dµ(x): X 1. Show that the set P (X) of all positive Radon measures of norm 1 (i.e. all Radon probability measures) on X is a weak* closed convex subset of M(X). 2. Show that the extreme points of P (X) are exactly Dirac measures δx with x 2 X. 3. The map D : x 2 X ! δx 2 Ext(P (X)) is a homeomorphism from topo- logical space (X; τ) onto the topological space (Ext(P (X)); τw∗ ). Proof of Part 1: Let fµαg be a net of positive Radon measure in P (X) such that µα ! µ 2 M(X) in weak* topology. We have Z Z fdµ = lim fdµα ≥ 0 X α X for all positive functions f in C(X). This shows that µ is a positive Radon measure. Since µα(X) = kµαk = 1 for all index α, we have Z Z µ(X) = 1dµ = lim 1dµα = lim µα(X) = 1: X α X α This shows that µ is a positive Radon measure with kµk = µ(X) = 1. Therefore P (X) is a weak* closed subset of M(X). Proof of Part 2: Let us fix a point x 2 X and let δx be the Dirac measure at x. Then δx is a Radon measure on X. Suppose µ = αµ1 +(1−α)µ2 for some 1 < α < 1 1 and probability Radon measures µ1 and µ2 in P (X). Since for arbitrary open set O with x2 = O, we have µ(O) = 0, this implies that µi(O) = 0 and thus µi = δx for i = 1; 2. This shows that δx is an extreme point in P (X) 3 Now we want to show that if µ is an extreme point of P (X), it must be a Dirac measure µ = δx for some point x. Suppose not. Then there exist two distinct points x and y in the support of µ, i.e. there exist two disjoint open sets x 2 U and y 2 V such that µ(U) > 0 and µ(V ) > 0. In this case, we have 0 < α = µ(U) < 1, µjU µjUc and µ1 = α and µ2 = (1−α) are positive Radon measures on X (see Homework 3, #1) with kµik = µi(X) = 1. It is easy to see that µ = αµ1 + (1 − α)µ2 is a convex combination of µ1 and µ2. Therefore, µ is not an extreme point, a contradiction. Proof of Part 3: It is known from the above discussion that D : x 2 X ! δx 2 Ext(P (X)) is a well-defined bijection. If a net xα ! x in X, then for every continuous function f 2 C(X), we have δxα (f) = f(xα) ! f(x) = δx(f): This shows that the map D is continuous from X onto Ext(P (X)). To show D−1 is continuous, we notice that for every closed subset F of X, F is compact and thus D(F ) is compact, and thus closed. It follows that if O is an open subset in X, then D(O) is open in Ext(P (X)). This implies that D−1 is also continuous. Therefore, D is a homeomorphism from X onto Ext(P (X))..
Load more

Recommended publications
  • Integral Representation of a Linear Functional on Function Spaces
    INTEGRAL REPRESENTATION OF LINEAR FUNCTIONALS ON FUNCTION SPACES MEHDI GHASEMI Abstract. Let A be a vector space of real valued functions on a non-empty set X and L : A −! R a linear functional. Given a σ-algebra A, of subsets of X, we present a necessary condition for L to be representable as an integral with respect to a measure µ on X such that elements of A are µ-measurable. This general result then is applied to the case where X carries a topological structure and A is a family of continuous functions and naturally A is the Borel structure of X. As an application, short solutions for the full and truncated K-moment problem are presented. An analogue of Riesz{Markov{Kakutani representation theorem is given where Cc(X) is replaced with whole C(X). Then we consider the case where A only consists of bounded functions and hence is equipped with sup-norm. 1. Introduction A positive linear functional on a function space A ⊆ RX is a linear map L : A −! R which assigns a non-negative real number to every function f 2 A that is globally non-negative over X. The celebrated Riesz{Markov{Kakutani repre- sentation theorem states that every positive functional on the space of continuous compactly supported functions over a locally compact Hausdorff space X, admits an integral representation with respect to a regular Borel measure on X. In symbols R L(f) = X f dµ, for all f 2 Cc(X). Riesz's original result [12] was proved in 1909, for the unit interval [0; 1].
    [Show full text]
  • Geometric Integration Theory Contents
    Steven G. Krantz Harold R. Parks Geometric Integration Theory Contents Preface v 1 Basics 1 1.1 Smooth Functions . 1 1.2Measures.............................. 6 1.2.1 Lebesgue Measure . 11 1.3Integration............................. 14 1.3.1 Measurable Functions . 14 1.3.2 The Integral . 17 1.3.3 Lebesgue Spaces . 23 1.3.4 Product Measures and the Fubini–Tonelli Theorem . 25 1.4 The Exterior Algebra . 27 1.5 The Hausdorff Distance and Steiner Symmetrization . 30 1.6 Borel and Suslin Sets . 41 2 Carath´eodory’s Construction and Lower-Dimensional Mea- sures 53 2.1 The Basic Definition . 53 2.1.1 Hausdorff Measure and Spherical Measure . 55 2.1.2 A Measure Based on Parallelepipeds . 57 2.1.3 Projections and Convexity . 57 2.1.4 Other Geometric Measures . 59 2.1.5 Summary . 61 2.2 The Densities of a Measure . 64 2.3 A One-Dimensional Example . 66 2.4 Carath´eodory’s Construction and Mappings . 67 2.5 The Concept of Hausdorff Dimension . 70 2.6 Some Cantor Set Examples . 73 i ii CONTENTS 2.6.1 Basic Examples . 73 2.6.2 Some Generalized Cantor Sets . 76 2.6.3 Cantor Sets in Higher Dimensions . 78 3 Invariant Measures and the Construction of Haar Measure 81 3.1 The Fundamental Theorem . 82 3.2 Haar Measure for the Orthogonal Group and the Grassmanian 90 3.2.1 Remarks on the Manifold Structure of G(N,M).... 94 4 Covering Theorems and the Differentiation of Integrals 97 4.1 Wiener’s Covering Lemma and its Variants .
    [Show full text]
  • On Stochastic Distributions and Currents
    NISSUNA UMANA INVESTIGAZIONE SI PUO DIMANDARE VERA SCIENZIA S’ESSA NON PASSA PER LE MATEMATICHE DIMOSTRAZIONI LEONARDO DA VINCI vol. 4 no. 3-4 2016 Mathematics and Mechanics of Complex Systems VINCENZO CAPASSO AND FRANCO FLANDOLI ON STOCHASTIC DISTRIBUTIONS AND CURRENTS msp MATHEMATICS AND MECHANICS OF COMPLEX SYSTEMS Vol. 4, No. 3-4, 2016 dx.doi.org/10.2140/memocs.2016.4.373 ∩ MM ON STOCHASTIC DISTRIBUTIONS AND CURRENTS VINCENZO CAPASSO AND FRANCO FLANDOLI Dedicated to Lucio Russo, on the occasion of his 70th birthday In many applications, it is of great importance to handle random closed sets of different (even though integer) Hausdorff dimensions, including local infor- mation about initial conditions and growth parameters. Following a standard approach in geometric measure theory, such sets may be described in terms of suitable measures. For a random closed set of lower dimension with respect to the environment space, the relevant measures induced by its realizations are sin- gular with respect to the Lebesgue measure, and so their usual Radon–Nikodym derivatives are zero almost everywhere. In this paper, how to cope with these difficulties has been suggested by introducing random generalized densities (dis- tributions) á la Dirac–Schwarz, for both the deterministic case and the stochastic case. For the last one, mean generalized densities are analyzed, and they have been related to densities of the expected values of the relevant measures. Ac- tually, distributions are a subclass of the larger class of currents; in the usual Euclidean space of dimension d, currents of any order k 2 f0; 1;:::; dg or k- currents may be introduced.
    [Show full text]
  • Invariant Radon Measures on Measured Lamination Space
    INVARIANT RADON MEASURES ON MEASURED LAMINATION SPACE URSULA HAMENSTADT¨ Abstract. Let S be an oriented surface of genus g ≥ 0 with m ≥ 0 punctures and 3g − 3+ m ≥ 2. We classify all Radon measures on the space of measured geodesic laminations which are invariant under the action of the mapping class group of S. 1. Introduction Let S be an oriented surface of finite type, i.e. S is a closed surface of genus g 0 from which m 0 points, so-called punctures, have been deleted. We assume that≥ 3g 3 + m 1,≥ i.e. that S is not a sphere with at most 3 punctures or a torus without− puncture.≥ In particular, the Euler characteristic of S is negative. Then the Teichm¨uller space (S) of S is the quotient of the space of all complete hyperbolic metrics of finite volumeT on S under the action of the group of diffeomorphisms of S which are isotopic to the identity. The mapping class group MCG(S) of all isotopy classes of orientation preserving diffeomorphisms of S acts properly discontinuously on (S) with quotient the moduli space Mod(S). T A geodesic lamination for a fixed choice of a complete hyperbolic metric of finite volume on S is a compact subset of S foliated into simple geodesics. A measured geodesic lamination is a geodesic lamination together with a transverse translation invariant measure. The space of all measured geodesic laminations on S, equipped with the weak∗-topology,ML is homeomorphic to S6g−7+2m (0, ) where S6g−7+2m is the 6g 7+2m-dimensional sphere.
    [Show full text]
  • Nonlinear Structures Determined by Measures on Banach Spaces Mémoires De La S
    MÉMOIRES DE LA S. M. F. K. DAVID ELWORTHY Nonlinear structures determined by measures on Banach spaces Mémoires de la S. M. F., tome 46 (1976), p. 121-130 <http://www.numdam.org/item?id=MSMF_1976__46__121_0> © Mémoires de la S. M. F., 1976, tous droits réservés. L’accès aux archives de la revue « Mémoires de la S. M. F. » (http://smf. emath.fr/Publications/Memoires/Presentation.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression 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/ Journees Geom. dimens. infinie [1975 - LYON ] 121 Bull. Soc. math. France, Memoire 46, 1976, p. 121 - 130. NONLINEAR STRUCTURES DETERMINED BY MEASURES ON BANACH SPACES By K. David ELWORTHY 0. INTRODUCTION. A. A Gaussian measure y on a separable Banach space E, together with the topolcT- gical vector space structure of E, determines a continuous linear injection i : H -> E, of a Hilbert space H, such that y is induced by the canonical cylinder set measure of H. Although the image of H has measure zero, nevertheless H plays a dominant role in both linear and nonlinear analysis involving y, [ 8] , [9], [10] . The most direct approach to obtaining measures on a Banach manifold M, related to its differential structure, requires a lot of extra structure on the manifold : for example a linear map i : H -> T M for each x in M, and even a subset M-^ of M which has the structure of a Hilbert manifold, [6] , [7].
    [Show full text]
  • Probability Measures on Metric Spaces
    Probability measures on metric spaces Onno van Gaans These are some loose notes supporting the first sessions of the seminar Stochastic Evolution Equations organized by Dr. Jan van Neerven at the Delft University of Technology during Winter 2002/2003. They contain less information than the common textbooks on the topic of the title. Their purpose is to present a brief selection of the theory that provides a basis for later study of stochastic evolution equations in Banach spaces. The notes aim at an audience that feels more at ease in analysis than in probability theory. The main focus is on Prokhorov's theorem, which serves both as an important tool for future use and as an illustration of techniques that play a role in the theory. The field of measures on topological spaces has the luxury of several excellent textbooks. The main source that has been used to prepare these notes is the book by Parthasarathy [6]. A clear exposition is also available in one of Bour- baki's volumes [2] and in [9, Section 3.2]. The theory on the Prokhorov metric is taken from Billingsley [1]. The additional references for standard facts on general measure theory and general topology have been Halmos [4] and Kelley [5]. Contents 1 Borel sets 2 2 Borel probability measures 3 3 Weak convergence of measures 6 4 The Prokhorov metric 9 5 Prokhorov's theorem 13 6 Riesz representation theorem 18 7 Riesz representation for non-compact spaces 21 8 Integrable functions on metric spaces 24 9 More properties of the space of probability measures 26 1 The distribution of a random variable in a Banach space X will be a probability measure on X.
    [Show full text]
  • Continuous Wavelet Transformation on Homogeneous Spaces
    Continuous Wavelet Transformation on Homogeneous Spaces Dissertation zur Erlangung des mathematisch-naturwissenschaftlichen Doktorgrades “Doctor rerum naturalium” der Georg-August-Universität zu Göttingen im Promotionsprogramm “Mathematical Sciences (Ph.D)” der Georg-August University School of Science (GAUSS) vorgelegt von Burkhard Blobel aus Freiberg/Sa. Göttingen, 2020 Betreuungsausschuss: Prof. Dr. Dorothea Bahns, Mathematisches Institut, Universität Göttingen Prof. Dr. Thomas Schick, Mathematisches Institut, Universität Göttingen Mitglieder der Prüfungskommission: Referent: Prof. Dr. Dorothea Bahns, Mathematisches Institut, Universität Göttingen Korreferent: Prof. Dr. Thomas Schick, Mathematisches Institut, Universität Göttingen Weitere Mitglieder der Prüfungskommission: Jun.-Prof. Dr. Madeleine Jotz Lean Mathematisches Institut, Universität Göttingen Prof. Dr. Ralf Meyer, Mathematisches Institut, Universität Göttingen Prof. Dr. Gerlind Plonka-Hoch, Institut für Numerische und Angewandte Mathematik, Universität Göttingen Prof. Dr. Dominic Schuhmacher Institut für Mathematische Stochastik, Universität Göttingen Tag der mündlichen Prüfung: 11.12.2020 Contents Notation and convention v 1 Introduction 1 2 Basics 7 2.1 Locally compact groups . .7 2.1.1 Topological groups . .7 2.1.2 Homogeneous spaces of locally compact groups . .9 2.1.3 Measures on locally compact spaces . 10 2.1.4 Haar measure . 13 2.1.5 Function spaces . 15 2.1.6 Quasi-invariant measures on the quotient space . 17 2.1.7 Standard Borel spaces . 22 2.2 Representation theory . 27 2.2.1 Unitary representation . 27 2.2.2 Induced representations . 30 2.2.3 Properties of induced representations . 34 2.3 The dual and the quasi-dual of a locally compact group . 35 2.3.1 Von Neumann algebras . 35 2.3.2 Direct integral of Hilbert spaces .
    [Show full text]
  • A Theory of Radon Measures on Locally Compact Spaces
    A THEORY OF RADON MEASURES ON LOCALLY COMPACT SPACES. By R. E. EDWARDS. 1. Introduction. In functional analysis one is often presented with the following situation: a locally compact space X is given, and along with it a certain topological vector space • of real functions defined on X; it is of importance to know the form of the most general continuous linear functional on ~. In many important cases, s is a superspace of the vector space ~ of all reul, continuous functions on X which vanish outside compact subsets of X, and the topology of s is such that if a sequence (]n) tends uniformly to zero and the ]n collectively vanish outside a fixed compact subset of X, then (/~) is convergent to zero in the sense of E. In this case the restriction to ~ of any continuous linear functional # on s has the property that #(/~)~0 whenever the se- quence (/~) converges to zero in the manner just described. It is therefore an important advance to determine all the linear functionals on (~ which are continuous in this eense. It is customary in some circles (the Bourbaki group, for example) to term such a functional # on ~ a "Radon measure on X". Any such functional can be written in many ways as the difference of two similar functionals, euch having the additional property of being positive in the sense that they assign a number >_ 0 to any function / satisfying ] (x) > 0 for all x E X. These latter functionals are termed "positive Radon measures on X", and it is to these that we may confine our attention.
    [Show full text]
  • Some Special Results of Measure Theory
    Some Special Results of Measure Theory By L. Le Cam1 U.C. Berkeley 1. Introduction The purpose of the present report is to record in writing some results of measure theory that are known to many people but do not seem to be in print, or at least seem to be difficult to find in the printed literature. The first result, originally proved by a consortium including R.M. Dudley, J. Feldman, D. Fremlin, C.C. Moore and R. Solovay in 1970 says something like this: Let X be compact with a Radon measure µ.Letf be a map from X to a metric space Y such that for every open set S ⊂ Y the inverse image, f −1(S) is Radon measurable. Then, if the cardinality of f(X)isnot outlandishly large, there is a subset X0 ⊂ X such that µ(X\X0)=0and f(X0) is separable. Precise definition of what outlandishly large means will be given below. The theorem may not appear very useful. However, after seeing it, one usually looks at empirical measures and processes in a different light. The theorem could be stated briefly as follows: A measurable image of a Radon measure in a complete metric space is Radon. Section 6 Theorem 7 gives an extension of the result where maps are replaced by Markov kernels. Sec- tion 8, Theorem 9, gives an extension to the case where the range space is paracompact instead of metric. The second part of the paper is an elaboration on certain classes of mea- sures that are limits in a suitable sense of “molecular” ones, that is measures carried by finite sets.
    [Show full text]
  • Radon Measures
    MAT 533, SPRING 2021, Stony Brook University REAL ANALYSIS II FOLLAND'S REAL ANALYSIS: CHAPTER 7 RADON MEASURES Christopher Bishop 1. Chapter 7: Radon Measures Chapter 7: Radon Measures 7.1 Positive linear functionals on Cc(X) 7.2 Regularity and approximation theorems 7.3 The dual of C0(X) 7.4* Products of Radon measures 7.5 Notes and References Chapter 7.1: Positive linear functionals X = locally compact Hausdorff space (LCH space) . Cc(X) = continuous functionals with compact support. Defn: A linear functional I on C0(X) is positive if I(f) ≥ 0 whenever f ≥ 0, Example: I(f) = f(x0) (point evaluation) Example: I(f) = R fdµ, where µ gives every compact set finite measure. We will show these are only examples. Prop. 7.1; If I is a positive linear functional on Cc(X), for each compact K ⊂ X there is a constant CK such that jI(f)j ≤ CLkfku for all f 2 Cc(X) such that supp(f) ⊂ K. Proof. It suffices to consider real-valued I. Given a compact K, choose φ 2 Cc(X; [0; 1]) such that φ = 1 on K (Urysohn's lemma). Then if supp(f) ⊂ K, jfj ≤ kfkuφ, or kfkφ − f > 0;; kfkφ + f > 0; so kfkuI(φ) − I)f) ≥ 0; kfkuI(φ) + I)f) ≥ 0: Thus jI(f)j ≤ I(φ)kfku: Defn: let µ be a Borel measure on X and E a Borel subset of X. µ is called outer regular on E if µ(E) = inffµ(U): U ⊃ E; U open g; and is inner regular on E if µ(E) = supfµ(K): K ⊂ E; K open g: Defn: if µ is outer and inner regular on all Borel sets, then it is called regular.
    [Show full text]
  • Representation of the Dirac Delta Function in C(R)
    Representation of the Dirac delta function in C(R∞) in terms of infinite-dimensional Lebesgue measures Gogi Pantsulaia∗ and Givi Giorgadze I.Vekua Institute of Applied Mathematics, Tbilisi - 0143, Georgian Republic e-mail: [email protected] Georgian Technical University, Tbilisi - 0175, Georgian Republic g.giorgadze Abstract: A representation of the Dirac delta function in C(R∞) in terms of infinite-dimensional Lebesgue measures in R∞ is obtained and some it’s properties are studied in this paper. MSC 2010 subject classifications: Primary 28xx; Secondary 28C10. Keywords and phrases: The Dirac delta function, infinite-dimensional Lebesgue measure. 1. Introduction The Dirac delta function(δ-function) was introduced by Paul Dirac at the end of the 1920s in an effort to create the mathematical tools for the development of quantum field theory. He referred to it as an improper functional in Dirac (1930). Later, in 1947, Laurent Schwartz gave it a more rigorous mathematical definition as a spatial linear functional on the space of test functions D (the set of all real-valued infinitely differentiable functions with compact support). Since the delta function is not really a function in the classical sense, one should not consider the value of the delta function at x. Hence, the domain of the delta function is D and its value for f ∈ D is f(0). Khuri (2004) studied some interesting applications of the delta function in statistics. The purpose of the present paper is an introduction of a concept of the Dirac delta function in the class of all continuous functions defined in the infinite- dimensional topological vector space of all real valued sequences R∞ equipped arXiv:1605.02723v2 [math.CA] 14 May 2016 with Tychonoff topology and a representation of this functional in terms of infinite-dimensional Lebesgue measures in R∞.
    [Show full text]
  • Real Analysis II, Winter 2018
    Real Analysis II, Winter 2018 From the Finnish original “Moderni reaalianalyysi”1 by Ilkka Holopainen adapted by Tuomas Hytönen February 22, 2018 1Version dated September 14, 2011 Contents 1 General theory of measure and integration 2 1.1 Measures . 2 1.11 Metric outer measures . 4 1.20 Regularity of measures, Radon measures . 7 1.31 Uniqueness of measures . 9 1.36 Extension of measures . 11 1.45 Product measure . 14 1.52 Fubini’s theorem . 16 2 Hausdorff measures 21 2.1 Basic properties of Hausdorff measures . 21 2.12 Hausdorff dimension . 24 2.17 Hausdorff measures on Rn ...................... 25 3 Compactness and convergence of Radon measures 30 3.1 Riesz representation theorem . 30 3.13 Weak convergence of measures . 35 3.17 Compactness of measures . 36 4 On the Hausdorff dimension of fractals 39 4.1 Mass distribution and Frostman’s lemma . 39 4.16 Self-similar fractals . 43 5 Differentiation of measures 52 5.1 Besicovitch and Vitali covering theorems . 52 1 Chapter 1 General theory of measure and integration 1.1 Measures Let X be a set and P(X) = fA : A ⊂ Xg its power set. Definition 1.2. A collection M ⊂ P (X) is a σ-algebra of X if 1. ? 2 M; 2. A 2 M ) Ac = X n A 2 M; S1 3. Ai 2 M, i 2 N ) i=1 Ai 2 M. Example 1.3. 1. P(X) is the largest σ-algebra of X; 2. f?;Xg is the smallest σ-algebra of X; 3. Leb(Rn) = the Lebesgue measurable subsets of Rn; 4.
    [Show full text]