The Riesz Representation Theorem for Positive Linear Functionals
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Vector Measures with Variation in a Banach Function Space
VECTOR MEASURES WITH VARIATION IN A BANACH FUNCTION SPACE O. BLASCO Departamento de An´alisis Matem´atico, Universidad de Valencia, Doctor Moliner, 50 E-46100 Burjassot (Valencia), Spain E-mail:[email protected] PABLO GREGORI Departament de Matem`atiques, Universitat Jaume I de Castell´o, Campus Riu Sec E-12071 Castell´o de la Plana (Castell´o), Spain E-mail:[email protected] Let E be a Banach function space and X be an arbitrary Banach space. Denote by E(X) the K¨othe-Bochner function space defined as the set of measurable functions f :Ω→ X such that the nonnegative functions fX :Ω→ [0, ∞) are in the lattice E. The notion of E-variation of a measure —which allows to recover the p- variation (for E = Lp), Φ-variation (for E = LΦ) and the general notion introduced by Gresky and Uhl— is introduced. The space of measures of bounded E-variation VE (X) is then studied. It is shown, amongother thingsand with some restriction ∗ ∗ of absolute continuity of the norms, that (E(X)) = VE (X ), that VE (X) can be identified with space of cone absolutely summingoperators from E into X and that E(X)=VE (X) if and only if X has the RNP property. 1. Introduction The concept of variation in the frame of vector measures has been fruitful in several areas of the functional analysis, such as the description of the duality of vector-valued function spaces such as certain K¨othe-Bochner function spaces (Gretsky and Uhl10, Dinculeanu7), the reformulation of operator ideals such as the cone absolutely summing operators (Blasco4), and the Hardy spaces of harmonic function (Blasco2,3). -
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]. -
The Representation Theorem of Persistence Revisited and Generalized
Journal of Applied and Computational Topology https://doi.org/10.1007/s41468-018-0015-3 The representation theorem of persistence revisited and generalized René Corbet1 · Michael Kerber1 Received: 9 November 2017 / Accepted: 15 June 2018 © The Author(s) 2018 Abstract The Representation Theorem by Zomorodian and Carlsson has been the starting point of the study of persistent homology under the lens of representation theory. In this work, we give a more accurate statement of the original theorem and provide a complete and self-contained proof. Furthermore, we generalize the statement from the case of linear sequences of R-modules to R-modules indexed over more general monoids. This generalization subsumes the Representation Theorem of multidimen- sional persistence as a special case. Keywords Computational topology · Persistence modules · Persistent homology · Representation theory Mathematics Subject Classifcation 06F25 · 16D90 · 16W50 · 55U99 · 68W30 1 Introduction Persistent homology, introduced by Edelsbrunner et al. (2002), is a multi-scale extension of classical homology theory. The idea is to track how homological fea- tures appear and disappear in a shape when the scale parameter is increasing. This data can be summarized by a barcode where each bar corresponds to a homology class that appears in the process and represents the range of scales where the class is present. The usefulness of this paradigm in the context of real-world data sets has led to the term topological data analysis; see the surveys (Carlsson 2009; Ghrist 2008; Edelsbrunner and Morozov 2012; Vejdemo-Johansson 2014; Kerber 2016) and textbooks (Edelsbrunner and Harer 2010; Oudot 2015) for various use cases. * René Corbet [email protected] Michael Kerber [email protected] 1 Institute of Geometry, Graz University of Technology, Kopernikusgasse 24, 8010 Graz, Austria Vol.:(0123456789)1 3 R. -
Representations of Direct Products of Finite Groups
Pacific Journal of Mathematics REPRESENTATIONS OF DIRECT PRODUCTS OF FINITE GROUPS BURTON I. FEIN Vol. 20, No. 1 September 1967 PACIFIC JOURNAL OF MATHEMATICS Vol. 20, No. 1, 1967 REPRESENTATIONS OF DIRECT PRODUCTS OF FINITE GROUPS BURTON FEIN Let G be a finite group and K an arbitrary field. We denote by K(G) the group algebra of G over K. Let G be the direct product of finite groups Gι and G2, G — G1 x G2, and let Mi be an irreducible iΓ(G;)-module, i— 1,2. In this paper we study the structure of Mί9 M2, the outer tensor product of Mi and M2. While Mi, M2 is not necessarily an irreducible K(G)~ module, we prove below that it is completely reducible and give criteria for it to be irreducible. These results are applied to the question of whether the tensor product of division algebras of a type arising from group representation theory is a division algebra. We call a division algebra D over K Z'-derivable if D ~ Hom^s) (Mf M) for some finite group G and irreducible ϋΓ(G)- module M. If B(K) is the Brauer group of K, the set BQ(K) of classes of central simple Z-algebras having division algebra components which are iΓ-derivable forms a subgroup of B(K). We show also that B0(K) has infinite index in B(K) if K is an algebraic number field which is not an abelian extension of the rationale. All iΓ(G)-modules considered are assumed to be unitary finite dimensional left if((τ)-modules. -
(Measure Theory for Dummies) UWEE Technical Report Number UWEETR-2006-0008
A Measure Theory Tutorial (Measure Theory for Dummies) Maya R. Gupta {gupta}@ee.washington.edu Dept of EE, University of Washington Seattle WA, 98195-2500 UWEE Technical Report Number UWEETR-2006-0008 May 2006 Department of Electrical Engineering University of Washington Box 352500 Seattle, Washington 98195-2500 PHN: (206) 543-2150 FAX: (206) 543-3842 URL: http://www.ee.washington.edu A Measure Theory Tutorial (Measure Theory for Dummies) Maya R. Gupta {gupta}@ee.washington.edu Dept of EE, University of Washington Seattle WA, 98195-2500 University of Washington, Dept. of EE, UWEETR-2006-0008 May 2006 Abstract This tutorial is an informal introduction to measure theory for people who are interested in reading papers that use measure theory. The tutorial assumes one has had at least a year of college-level calculus, some graduate level exposure to random processes, and familiarity with terms like “closed” and “open.” The focus is on the terms and ideas relevant to applied probability and information theory. There are no proofs and no exercises. Measure theory is a bit like grammar, many people communicate clearly without worrying about all the details, but the details do exist and for good reasons. There are a number of great texts that do measure theory justice. This is not one of them. Rather this is a hack way to get the basic ideas down so you can read through research papers and follow what’s going on. Hopefully, you’ll get curious and excited enough about the details to check out some of the references for a deeper understanding. -
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. -
Some Undecidability Results Concerning Radon Measures by R
transactions of the american mathematical society Volume 259, Number 1, May 1980 SOME UNDECIDABILITY RESULTS CONCERNING RADON MEASURES BY R. J. GARDNER AND W. F. PFEFFER Abstract. We show that in metalindelöf spaces certain questions about Radon measures cannot be decided within the Zermelo-Fraenkel set theory, including the axiom of choice. 0. Introduction. All spaces in this paper will be Hausdorff. Let A be a space. By § and Q we shall denote, respectively, the families of all open and compact subsets of X. The Borel o-algebra in X, i.e., the smallest a-algebra in X containing ê, will be denoted by <®. The elements of 9> are called Borel sets. A Borel measure in A' is a measure p on % such that each x G X has a neighborhood U G % with p( U) < + 00. A Borel measure p in X is called (i) Radon if p(B) = sup{ p(C): C G G, C c F} for each F G % ; (ii) regular if p(B) = inî{ p(G): G G @,B c G} for each fieS. A Radon space is a space X in which each finite Borel measure is Radon. The present paper will be divided into two parts, each dealing with one of the following questions. (a) Let A' be a locally compact, hereditarily metalindelöf space with the counta- ble chain condition. Is X a Radon space? (ß) Let A' be a metalindelöf space, and let p be a a-finite Radon measure in X. Is pa. regular measure? We shall show that neither question can be decided within the usual axioms of set theory. -
Representation Theory of Finite Groups
Representation Theory of Finite Groups Benjamin Steinberg School of Mathematics and Statistics Carleton University [email protected] December 15, 2009 Preface This book arose out of course notes for a fourth year undergraduate/first year graduate course that I taught at Carleton University. The goal was to present group representation theory at a level that is accessible to students who have not yet studied module theory and who are unfamiliar with tensor products. For this reason, the Wedderburn theory of semisimple algebras is completely avoided. Instead, I have opted for a Fourier analysis approach. This sort of approach is normally taken in books with a more analytic flavor; such books, however, invariably contain material on representations of com- pact groups, something that I would also consider beyond the scope of an undergraduate text. So here I have done my best to blend the analytic and the algebraic viewpoints in order to keep things accessible. For example, Frobenius reciprocity is treated from a character point of view to evade use of the tensor product. The only background required for this book is a basic knowledge of linear algebra and group theory, as well as familiarity with the definition of a ring. The proof of Burnside's theorem makes use of a small amount of Galois theory (up to the fundamental theorem) and so should be skipped if used in a course for which Galois theory is not a prerequisite. Many things are proved in more detail than one would normally expect in a textbook; this was done to make things easier on undergraduates trying to learn what is usually considered graduate level material. -
The Algebraic Theory of Semigroups the Algebraic Theory of Semigroups
http://dx.doi.org/10.1090/surv/007.1 The Algebraic Theory of Semigroups The Algebraic Theory of Semigroups A. H. Clifford G. B. Preston American Mathematical Society Providence, Rhode Island 2000 Mathematics Subject Classification. Primary 20-XX. International Standard Serial Number 0076-5376 International Standard Book Number 0-8218-0271-2 Library of Congress Catalog Card Number: 61-15686 Copying and reprinting. Material in this book may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permissionQams.org. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) © Copyright 1961 by the American Mathematical Society. All rights reserved. Printed in the United States of America. Second Edition, 1964 Reprinted with corrections, 1977. The American Mathematical Society retains all rights except those granted to the United States Government. -
Chapter 2. Functions of Bounded Variation §1. Monotone Functions
Chapter 2. Functions of Bounded Variation §1. Monotone Functions Let I be an interval in IR. A function f : I → IR is said to be increasing (strictly increasing) if f(x) ≤ f(y)(f(x) < f(y)) whenever x, y ∈ I and x < y. A function f : I → IR is said to be decreasing (strictly decreasing) if f(x) ≥ f(y)(f(x) > f(y)) whenever x, y ∈ I and x < y. A function f : I → IR is said to be monotone if f is either increasing or decreasing. In this section we will show that a monotone function is differentiable almost every- where. For this purpose, we first establish Vitali covering lemma. Let E be a subset of IR, and let Γ be a collection of closed intervals in IR. We say that Γ covers E in the sense of Vitali if for each δ > 0 and each x ∈ E, there exists an interval I ∈ Γ such that x ∈ I and `(I) < δ. Theorem 1.1. Let E be a subset of IR with λ∗(E) < ∞ and Γ a collection of closed intervals that cover E in the sense of Vitali. Then, for given ε > 0, there exists a finite disjoint collection {I1,...,IN } of intervals in Γ such that ∗ N λ E \ ∪n=1In < ε. Proof. Let G be an open set containing E such that λ(G) < ∞. Since Γ is a Vitali covering of E, we may assume that each I ∈ Γ is contained in G. We choose a sequence (In)n=1,2,... of disjoint intervals from Γ recursively as follows. -
Section 3.5 of Folland’S Text, Which Covers Functions of Bounded Variation on the Real Line and Related Topics
REAL ANALYSIS LECTURE NOTES: 3.5 FUNCTIONS OF BOUNDED VARIATION CHRISTOPHER HEIL 3.5.1 Definition and Basic Properties of Functions of Bounded Variation We will expand on the first part of Section 3.5 of Folland's text, which covers functions of bounded variation on the real line and related topics. We begin with functions defined on finite closed intervals in R (note that Folland's ap- proach and notation is slightly different, as he begins with functions defined on R and uses TF (x) instead of our V [f; a; b]). Definition 1. Let f : [a; b] ! C be given. Given any finite partition Γ = fa = x0 < · · · < xn = bg of [a; b], set n SΓ = jf(xi) − f(xi−1)j: i=1 X The variation of f over [a; b] is V [f; a; b] = sup SΓ : Γ is a partition of [a; b] : The function f has bounded variation on [a; b] if V [f; a; b] < 1. We set BV[a; b] = f : [a; b] ! C : f has bounded variation on [a; b] : Note that in this definition we are considering f to be defined at all poin ts, and not just to be an equivalence class of functions that are equal a.e. The idea of the variation of f is that is represents the total vertical distance traveled by a particle that moves along the graph of f from (a; f(a)) to (b; f(b)). Exercise 2. (a) Show that if f : [a; b] ! C, then V [f; a; b] ≥ jf(b) − f(a)j. -
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.