Vector Measures, Integration and Applications
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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). -
Combinatorics in Banach Space Theory Lecture 2
Combinatorics in Banach space theory Lecture 2 Now, we will derive the promised corollaries from Rosenthal's lemma. First, following [Ros70], we will show how this lemma implies Nikod´ym's uniform boundedness principle for bounded vector measures. Before doing this we need to recall some definitions. Definition 2.4. Let F be a set algebra. By a partition of a given set E 2 F we mean Sk a finite collection fE1;:::;Ekg of pairwise disjoint members of F such that j=1Ej = E. We denote Π(E) the set of all partitions of E. Let also X be a Banach space and µ: F ! X be a finitely additive set function (called a vector measure). The variation of µ is a function jµj: F ! [0; 1] given by ( ) X jµj(E) = sup kµ(A)k: π 2 Π(E) : A2π The semivariation of µ is a function kµk: F ! [0; 1] given by ∗ ∗ kµk = sup jx µj(E): x 2 BX∗ : By a straightforward calculation, one may check that the variation jµj is always finitely additive, whereas the semivariation kµk is monotone and finitely subadditive. Of course, we always have kµk 6 jµj. Moreover, since we know how the functionals on the scalar space (R or C) look like, it is easily seen that for scalar-valued measures the notions of variation and semivariation coincide. By saying that a vector measure µ: F ! X is bounded we mean that kµk is finitely valued. This is in turn equivalent to saying that the range of µ is a bounded subset of X. -
Banach Space Properties of V of a Vector Measure
proceedings of the american mathematical society Volume 123, Number 12. December 1995 BANACHSPACE PROPERTIES OF V OF A VECTOR MEASURE GUILLERMO P. CURBERA (Communicated by Dale Alspach) Abstract. We consider the space L'(i>) of real functions which are integrable with respect to a measure v with values in a Banach space X . We study type and cotype for Lx{v). We study conditions on the measure v and the Banach space X that imply that Ll(v) is a Hilbert space, or has the Dunford-Pettis property. We also consider weak convergence in Lx(v). 1. Introduction Given a vector measure v with values in a Banach space X, Lx(v) denotes the space of (classes of) real functions which are integrable with respect to v in the sense of Bartle, Dunford and Schwartz [BDS] and Lewis [L-l]. This space has been studied by Kluvanek and Knowles [KK], Thomas [T] and Okada [O]. It is an order continuous Banach lattice with weak unit. In [C-l, Theorem 8] we have identified the class of spaces Lx(v) , showing that every order continuous Banach lattice with weak unit can be obtained, order isometrically, as L1 of a suitable vector measure. A natural question arises: what is the relation between, on the one hand, the properties of the Banach space X and the measure v, and, on the other hand, the properties of the resulting space Lx(v) . The complexity of the situ- ation is shown by the following example: the measures, defined over Lebesgue measurable sets of [0,1], vx(A) = m(A) £ R, v2(A) = Xa e £'([0, 1]) and v3 — (]"Ar„(t)dt) £ Co, where rn are the Rademacher functions, generate, order isometrically, the same space, namely 7J([0, 1]). -
FLOWS of MEASURES GENERATED by VECTOR FIELDS 1. Introduction Every Sufficiently Smooth and Bounded Vector Field V in the Euclide
FLOWS OF MEASURES GENERATED BY VECTOR FIELDS EMANUELE PAOLINI AND EUGENE STEPANOV Abstract. The scope of the paper is twofold. We show that for a large class of measurable vector fields in the sense of N. Weaver (i.e. derivations over the algebra of Lipschitz functions), called in the paper laminated, the notion of integral curves may be naturally defined and characterized (when appropriate) by an ODE. We further show, that for such vector fields the notion of a flow of the given positive Borel measure similar to the classical one generated by a smooth vector field (in a space with smooth structure) may be defined in a reasonable way, so that the measure “flows along” the appropriately understood integral curves of the given vector field and the classical continuity equation is satisfied in the weak sense. 1. Introduction Every sufficiently smooth and bounded vector field V in the Euclidean space E = Rn is well-known to generate a canonical flow of a given finite Borel measure t + t µ in E by setting µt := ϕ µ, t R , where ϕ (x) := θ(t), θ standing for the V # ∈ V unique solution to the differential equation (1.1) θ˙(t)= V (θ(t)) satisfying the initial condition θ(0) = x. Such a flow satisfies the continuity equation ∂µ (1.2) t + div v µ =0 ∂t t t + in the weak sense in E R , where vt : E E is some velocity field (of course, non unique for the given×V , but in particular,→ this equation is satisfied in this case with vt := V ). -
Locally Solid Riesz Spaces with Applications to Economics / Charalambos D
http://dx.doi.org/10.1090/surv/105 alambos D. Alipr Lie University \ Burkinshaw na University-Purdue EDITORIAL COMMITTEE Jerry L. Bona Michael P. Loss Peter S. Landweber, Chair Tudor Stefan Ratiu J. T. Stafford 2000 Mathematics Subject Classification. Primary 46A40, 46B40, 47B60, 47B65, 91B50; Secondary 28A33. Selected excerpts in this Second Edition are reprinted with the permissions of Cambridge University Press, the Canadian Mathematical Bulletin, Elsevier Science/Academic Press, and the Illinois Journal of Mathematics. For additional information and updates on this book, visit www.ams.org/bookpages/surv-105 Library of Congress Cataloging-in-Publication Data Aliprantis, Charalambos D. Locally solid Riesz spaces with applications to economics / Charalambos D. Aliprantis, Owen Burkinshaw.—2nd ed. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 105) Rev. ed. of: Locally solid Riesz spaces. 1978. Includes bibliographical references and index. ISBN 0-8218-3408-8 (alk. paper) 1. Riesz spaces. 2. Economics, Mathematical. I. Burkinshaw, Owen. II. Aliprantis, Char alambos D. III. Locally solid Riesz spaces. IV. Title. V. Mathematical surveys and mono graphs ; no. 105. QA322 .A39 2003 bib'.73—dc22 2003057948 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. -
Algebras and Orthogonal Vector Measures
proceedings of the american mathematical society Volume 110, Number 3, November 1990 STATES ON IT*-ALGEBRAS AND ORTHOGONAL VECTOR MEASURES JAN HAMHALTER (Communicated by William D. Sudderth) Abstract. We show that every state on a If*-algebra sf without type I2 direct summand is induced by an orthogonal vector measure on sf . This result may find an application in quantum stochastics [1, 7]. Particularly, it allows us to find a simple formula for the transition probability between two states on sf [3, 8], 1. Preliminaries Here we fix some notation and recall basic facts about Hilbert-Schmidt op- erators (for details, see [9]). Let H be a complex Hilbert space and let 3§'HA) denote the set of all linear bounded operators on H . An operator A e 33(H) is said to be a trace class operator if the series ^2aeI(Aea, ea) is absolutely convergent for an orthonormal basis (ea)ae! of //. In this case the sum J2a€¡(Aea, ea) does not depend on the choice of the basis (ea)a€¡ and is called a trace of A (abbreviated Tr,4). Let us denote by CX(H) the set of all trace class operators on //. Then Tr AB = Tr BA whenever the product AB belongs to CX(H). Suppose that A e 33(H). Then A is called a Hilbert-Schmidt operator if IMII2 = {X3„e/II^JI } < 00 for some (and therefore for any) orthonor- mal basis (en)n€, of //. Let us denote by C2(H) the set of all Hilbert- Schmidt operators on //. Thus, C2(H) is a two-sided ideal in 33(H), and CX(H) c C2(H). -
Boolean Valued Analysis of Order Bounded Operators
BOOLEAN VALUED ANALYSIS OF ORDER BOUNDED OPERATORS A. G. Kusraev and S. S. Kutateladze Abstract This is a survey of some recent applications of Boolean valued models of set theory to order bounded operators in vector lattices. Key words: Boolean valued model, transfer principle, descent, ascent, order bounded operator, disjointness, band preserving operator, Maharam operator. Introduction The term Boolean valued analysis signifies the technique of studying proper- ties of an arbitrary mathematical object by comparison between its representa- tions in two different set-theoretic models whose construction utilizes principally distinct Boolean algebras. As these models, we usually take the classical Canto- rian paradise in the shape of the von Neumann universe and a specially-trimmed Boolean valued universe in which the conventional set-theoretic concepts and propositions acquire bizarre interpretations. Use of two models for studying a single object is a family feature of the so-called nonstandard methods of analy- sis. For this reason, Boolean valued analysis means an instance of nonstandard analysis in common parlance. Proliferation of Boolean valued analysis stems from the celebrated achieve- ment of P. J. Cohen who proved in the beginning of the 1960s that the negation of the continuum hypothesis, CH, is consistent with the axioms of Zermelo– Fraenkel set theory, ZFC. This result by Cohen, together with the consistency of CH with ZFC established earlier by K. G¨odel, proves that CH is independent of the conventional axioms of ZFC. The first application of Boolean valued models to functional analysis were given by E. I. Gordon for Dedekind complete vector lattices and positive op- erators in [22]–[24] and G. -
On Universally Complete Riesz Spaces
Pacific Journal of Mathematics ON UNIVERSALLY COMPLETE RIESZ SPACES CHARALAMBOS D. ALIPRANTIS AND OWEN SIDNEY BURKINSHAW Vol. 71, No. 1 November 1977 FACIFIC JOURNAL OF MATHEMATICS Vol. 71, No. 1, 1977 ON UNIVERSALLY COMPLETE RIESZ SPACES C. D. ALIPRANTIS AND 0. BURKINSHAW In Math. Proc. Cambridge Philos. Soc. (1975), D. H. Fremlin studied the structure of the locally solid topologies on inex- tensible Riesz spaces. He subsequently conjectured that his results should hold true for σ-universally complete Riesz spaces. In this paper we prove that indeed Fremlin's results can be generalized to σ-universally complete Riesz spaces and at the same time establish a number of new properties. Ex- amples of Archimedean universally complete Riesz spaces which are not Dedekind complete are also given. 1* Preliminaries* For notation and terminology concerning Riesz spaces not explained below we refer the reader to [10]* A Riesz space L is said to be universally complete if the supremum of every disjoint system of L+ exists in L [10, Definition 47.3, p. 323]; see also [12, p. 140], Similarly a Riesz space L is said to be a-univer sally complete if the supremum of every disjoint sequence of L+ exists in L. An inextensible Riesz space is a Dedekind complete and universally complete Riesz space [7, Definition (a), p. 72], In the terminology of [12] an inextensible Riesz space is called an extended Dedekind complete Riesz space; see [12, pp. 140-144]. Every band in a universally complete Riesz space is in its own right a universally complete Riesz space. -
Continuity Properties of Vector Measures
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 86. 268-280 (1982) Continuity Properties of Vector Measures CECILIA H. BROOK Department of Mathematical Sciences, Northern Illinois Unirersity, DeKalb, Illinois 60115 Submitted b! G.-C Rota 1. INTRODUCTION This paper is a sequel to [ I 1. Using projections in the universal measure space of W. H. Graves [ 6 1, we study continuity and orthogonality properties of vector measures. To each strongly countably additive measure 4 on an algebra .d with range in a complete locally convex space we associate a projection P,, which may be thought of as the support of @ and as a projection-valued measure. Our main result is that, as measures, # and P, are mutually continuous. It follows that measures may be compared by comparing their projections. The projection P, induces a decomposition of vector measures which turns out to be Traynor’s general Lebesgue decomposition [8) relative to qs. Those measures $ for which algebraic d-continuity is equivalent to $- continuity are characterized in terms of projections, while $-continuity is shown to be equivalent to &order-continuity whenever -d is a u-algebra. Using projections associated with nonnegative measures, we approximate every 4 uniformly on the algebra ,d by vector measures which have control measures. Last, we interpret our results algebraically. Under the equivalence relation of mutual continuity, and with the order induced by continuity, the collection of strongly countably additive measures on .d which take values in complete spaces forms a complete Boolean algebra 1rV, which is isomorphic to the Boolean algebra of projections in the universal measure space. -
THE RANGE of a VECTOR MEASURE the Purpose of This Note Is to Prove That the Range of a Countably Additive Finite Measure with Va
THE RANGE OF A VECTOR MEASURE PAUL R. HALMOS The purpose of this note is to prove that the range of a countably additive finite measure with values in a finite-dimensional real vector space is closed and, in the non-atomic case, convex. These results were first proved (in 1940) by A. Liapounoff.1 In 1945 K. R. Buch (independently) proved the part of the statement concerning closure for non-negative measures of dimension one or two.2 In 1947 I offered a proof of Buch's results which, however, was correct in the one- dimensional case only.8 In this paper I present a simplified proof of LiapounofPs results. Let X be any set and let S be a <r-field of subsets of X (called the measurable set of X). A measure /i (or, more precisely, an iV-dimen- sional measure (#i, • • • , fix)) is a (bounded) countably additive function of the sets of S with values in iV-dimensional, real vector space (in which the "length" |&| + • • • +|£JV| of a vector ? = (?i» • • • » &v) is denoted by |£|). The measure (/xi, • • • /*#) is non-negative if fXi(E) ^0 for every ££S and i«= 1, • • • , N. For a numerical (one-dimensional) measure /x0, Mo*0E) will denote the total variation of /xo on £; in general, if /x = (jiu • • • , Mtf), M* will denote the non-negative measure (jit!*, • • • , /$). The length |JU*| =Mi*+ • • • +M#* is always a non-negative numerical measure.4 Presented to the Society, September 3,1947 ; received by the editors July 10,1947. 1 Sur les fonctions-vecteurs complètement additives, Bull. -
Kuelbs–Steadman Spaces for Banach Space-Valued Measures
mathematics Article Kuelbs–Steadman Spaces for Banach Space-Valued Measures Antonio Boccuto 1,*,† , Bipan Hazarika 2,† and Hemanta Kalita 3,† 1 Department of Mathematics and Computer Sciences, University of Perugia, via Vanvitelli, 1 I-06123 Perugia, Italy 2 Department of Mathematics, Gauhati University, Guwahati 781014, Assam, India; [email protected] 3 Department of Mathematics, Patkai Christian College (Autonomous), Dimapur, Patkai 797103, Nagaland, India; [email protected] * Correspondence: [email protected] † These authors contributed equally to this work. Received: 2 June 2020; Accepted: 15 June 2020; Published: 19 June 2020 Abstract: We introduce Kuelbs–Steadman-type spaces (KSp spaces) for real-valued functions, with respect to countably additive measures, taking values in Banach spaces. We investigate the main properties and embeddings of Lq-type spaces into KSp spaces, considering both the norm associated with the norm convergence of the involved integrals and that related to the weak convergence of the integrals. Keywords: Kuelbs–Steadman space; Henstock–Kurzweil integrable function; vector measure; dense embedding; completely continuous embedding; Köthe space; Banach lattice MSC: Primary 28B05; 46G10; Secondary 46B03; 46B25; 46B40 1. Introduction Kuelbs–Steadman spaces have been the subject of many recent studies (see, e.g., [1–3] and the references therein). The investigation of such spaces arises from the idea to consider the L1 spaces as embedded in a larger Hilbert space with a smaller norm and containing in a certain sense the Henstock–Kurzweil integrable functions. This allows giving several applications to functional analysis and other branches of mathematics, for instance Gaussian measures (see also [4]), convolution operators, Fourier transforms, Feynman integrals, quantum mechanics, differential equations, and Markov chains (see also [1–3]). -
Version of 3.1.15 Chapter 56 Choice and Determinacy
Version of 3.1.15 Chapter 56 Choice and determinacy Nearly everyone reading this book will have been taking the axiom of choice for granted nearly all the time. This is the home territory of twentieth-century abstract analysis, and the one in which the great majority of the results have been developed. But I hope that everyone is aware that there are other ways of doing things. In this chapter I want to explore what seem to me to be the most interesting alternatives. In one sense they are minor variations on the standard approach, since I keep strictly to ideas expressible within the framework of Zermelo-Fraenkel set theory; but in other ways they are dramatic enough to rearrange our prejudices. The arguments I will present in this chapter are mostly not especially difficult by the standards of this volume, but they do depend on intuitions for which familiar results which are likely to remain valid under the new rules being considered. Let me say straight away that the real aim of the chapter is §567, on the axiom of determinacy. The significance of this axiom is that it is (so far) the most striking rival to the axiom of choice, in that it leads us quickly to a large number of propositions directly contradicting familiar theorems; for instance, every subset of the real line is now Lebesgue measurable (567G). But we need also to know which theorems are still true, and the first six sections of the chapter are devoted to a discussion of what can be done in ZF alone (§§561-565) and with countable or dependent choice (§566).