Weak Convergence of Measures
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Distinguished Property in Tensor Products and Weak* Dual Spaces
axioms Article Distinguished Property in Tensor Products and Weak* Dual Spaces Salvador López-Alfonso 1 , Manuel López-Pellicer 2,* and Santiago Moll-López 3 1 Department of Architectural Constructions, Universitat Politècnica de València, 46022 Valencia, Spain; [email protected] 2 Emeritus and IUMPA, Universitat Politècnica de València, 46022 Valencia, Spain 3 Department of Applied Mathematics, Universitat Politècnica de València, 46022 Valencia, Spain; [email protected] * Correspondence: [email protected] 0 Abstract: A local convex space E is said to be distinguished if its strong dual Eb has the topology 0 0 0 0 b(E , (Eb) ), i.e., if Eb is barrelled. The distinguished property of the local convex space Cp(X) of real- valued functions on a Tychonoff space X, equipped with the pointwise topology on X, has recently aroused great interest among analysts and Cp-theorists, obtaining very interesting properties and nice characterizations. For instance, it has recently been obtained that a space Cp(X) is distinguished if and only if any function f 2 RX belongs to the pointwise closure of a pointwise bounded set in C(X). The extensively studied distinguished properties in the injective tensor products Cp(X) ⊗# E and in Cp(X, E) contrasts with the few distinguished properties of injective tensor products related to the dual space Lp(X) of Cp(X) endowed with the weak* topology, as well as to the weak* dual of Cp(X, E). To partially fill this gap, some distinguished properties in the injective tensor product space Lp(X) ⊗# E are presented and a characterization of the distinguished property of the weak* dual of Cp(X, E) for wide classes of spaces X and E is provided. -
Weak Convergence of Probability Measures Revisited
WEAK CONVERGENCE OF PROBABILITY MEASURES REVISITED Cabriella saltnetti' Roger J-B wetse April 1987 W P-87-30 This research was supported in part by MPI, Projects: "Calcolo Stocastico e Sistemi Dinamici Statistici" and "Modelli Probabilistic" 1984, and by the National Science Foundation. Working Papers are interim reports on work of the lnternational Institute for Applied Systems Analysis and have received only limited review. Views or opinions expressed herein do not necessarily represent those of the Institute or of its National Member Organizations. INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSlS A-2361 Laxenburg, Austria FOREWORD The modeling of stochastic processes is a fundamental tool in the study of models in- volving uncertainty, a major topic at SDS. A number of classical convergence results (and extensions) for probability measures are derived by relying on new tools that are particularly useful in stochastic optimization and extremal statistics. Alexander B. Kurzhanski Chairman System and Decision Sciences Program ABSTRACT The hypo-convergence of upper semicontinuous functions provides a natural frame- work for the study of the convergence of probability measures. This approach also yields some further characterizations of weak convergence and tightness. CONTENTS 1 About Continuity and Measurability 2 Convergence of Sets and Semicontinuous Functions 3 SC-Measures and SGPrerneasures on 7 (E) 4 Hypo-Limits of SGMeasures and Tightness 5 Tightness and Equi-Semicontinuity Acknowledgment Appendix A Appendix B References - vii - WEAK CONVERGENCE OF PROBABILITY MEASURES REVISITED Gabriella salinettil and Roger J-B wetsL luniversiti "La Sapienza", 00185 Roma, Italy. 2~niversityof California, Davis, CA 95616. 1. ABOUT CONTINUITY AND MEASURABILITY A probabilistic structure - a space of possible events, a sigma-field of (observable) subcollections of events, and a probability measure defined on this sigma-field - does not have a built-in topological structure. -
Arxiv:1009.3824V2 [Math.OC] 7 Feb 2012 E Fpstv Nees Let Integers
OPTIMIZATION AND CONVERGENCE OF OBSERVATION CHANNELS IN STOCHASTIC CONTROL SERDAR YUKSEL¨ AND TAMAS´ LINDER Abstract. This paper studies the optimization of observation channels (stochastic kernels) in partially observed stochastic control problems. In particular, existence and continuity properties are investigated mostly (but not exclusively) concentrating on the single-stage case. Continuity properties of the optimal cost in channels are explored under total variation, setwise convergence, and weak convergence. Sufficient conditions for compactness of a class of channels under total variation and setwise convergence are presented and applications to quantization are explored. Key words. Stochastic control, information theory, observation channels, optimization, quan- tization AMS subject classifications. 15A15, 15A09, 15A23 1. Introduction. In stochastic control, one is often concerned with the following problem: Given a dynamical system, an observation channel (stochastic kernel), a cost function, and an action set, when does there exist an optimal policy, and what is an optimal control policy? The theory for such problems is advanced, and practically significant, spanning a wide variety of applications in engineering, economics, and natural sciences. In this paper, we are interested in a dual problem with the following questions to be explored: Given a dynamical system, a cost function, an action set, and a set of observation channels, does there exist an optimal observation channel? What is the right convergence notion for continuity in such observation channels for optimization purposes? The answers to these questions may provide useful tools for characterizing an optimal observation channel subject to constraints. We start with the probabilistic setup of the problem. Let X ⊂ Rn, be a Borel set in which elements of a controlled Markov process {Xt, t ∈ Z+} live. -
Arxiv:2102.05840V2 [Math.PR]
SEQUENTIAL CONVERGENCE ON THE SPACE OF BOREL MEASURES LIANGANG MA Abstract We study equivalent descriptions of the vague, weak, setwise and total- variation (TV) convergence of sequences of Borel measures on metrizable and non-metrizable topological spaces in this work. On metrizable spaces, we give some equivalent conditions on the vague convergence of sequences of measures following Kallenberg, and some equivalent conditions on the TV convergence of sequences of measures following Feinberg-Kasyanov-Zgurovsky. There is usually some hierarchy structure on the equivalent descriptions of convergence in different modes, but not always. On non-metrizable spaces, we give examples to show that these conditions are seldom enough to guarantee any convergence of sequences of measures. There are some remarks on the attainability of the TV distance and more modes of sequential convergence at the end of the work. 1. Introduction Let X be a topological space with its Borel σ-algebra B. Consider the collection M˜ (X) of all the Borel measures on (X, B). When we consider the regularity of some mapping f : M˜ (X) → Y with Y being a topological space, some topology or even metric is necessary on the space M˜ (X) of Borel measures. Various notions of topology and metric grow out of arXiv:2102.05840v2 [math.PR] 28 Apr 2021 different situations on the space M˜ (X) in due course to deal with the corresponding concerns of regularity. In those topology and metric endowed on M˜ (X), it has been recognized that the vague, weak, setwise topology as well as the total-variation (TV) metric are highlighted notions on the topological and metric description of M˜ (X) in various circumstances, refer to [Kal, GR, Wul]. -
Statistics and Probability Letters a Note on Vague Convergence Of
Statistics and Probability Letters 153 (2019) 180–186 Contents lists available at ScienceDirect Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro A note on vague convergence of measures ∗ Bojan Basrak, Hrvoje Planini¢ Department of Mathematics, Faculty of Science, University of Zagreb, Bijeni£ka 30, Zagreb, Croatia article info a b s t r a c t Article history: We propose a new approach to vague convergence of measures based on the general Received 17 April 2019 theory of boundedness due to Hu (1966). The article explains how this connects and Accepted 6 June 2019 unifies several frequently used types of vague convergence from the literature. Such Available online 20 June 2019 an approach allows one to translate already developed results from one type of vague MSC: convergence to another. We further analyze the corresponding notion of vague topology primary 28A33 and give a new and useful characterization of convergence in distribution of random secondary 60G57 measures in this topology. 60G70 ' 2019 Elsevier B.V. All rights reserved. Keywords: Boundedly finite measures Vague convergence w#–convergence Random measures Convergence in distribution Lipschitz continuous functions 1. Introduction Let X be a Polish space, i.e. separable topological space which is metrizable by a complete metric. Denote by B(X) the corresponding Borel σ –field and choose a subfamily Bb(X) ⊆ B(X) of sets, called bounded (Borel) sets of X. When there is no fear of confusion, we will simply write B and Bb. A Borel measure µ on X is said to be locally (or boundedly) finite if µ(B) < 1 for all B 2 Bb. -
HISTORICAL NOTES Chapter 1:. the Idea of Topologizing the Set of Continuous Functions from One Topological Space Into Another To
HISTORICAL NOTES Chapter 1:. The idea of topologizing the set of continuous functions from one topological space into another topological space arose from the notions of pointwise and uniform convergence of sequences of functions. Apparently the work of Ascoli [1883], [1889] and Hadamard [1898] marked the beginning of function space theory. The topology of pointwise convergence and the topology of uniform convergence are among the first function space topologies considered in the early years of general topology. The I supremum metric topology was studied in Frechet [19061. The paper of Tychonoff [1935] showed that the (Tychonoff) product on the set RX is nothing but the topology of pointwise convergence. In 1945, Fox [19451 defined the compact-open topology. Shortly thereafter, Arens [1946] studied this topology, which he called k-topology. Among other things which Arens proved was the compact-open topology version of Theorem 1.2.3. Set-open topologies in a more general setting were studied by Arens and Dugundji [1951] in connection with the concepts of admissible and proper topologies. Theorem 1.2.5 is due to Jackson [1952], and Example 1.2.7 can be found in Dugundji [1968]. Chapter 1:. Admissible [i.e., conjoining) topologies were introduced by Arens [1946] and splitting (i.e., proper) topologies were studied by Arens and Dugundji [1951], where they proved Theorem 2.5.3. Proofs of Theorem 2.5.2 and Corollary 2.5.4.a can be found in Fox [1945]. Corollary 2.5.7 is apparently due to Jackson [1952]; and Morita [1956] proved Corollary 2.5.8. -
Lecture 20: Weak Convergence (PDF)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 20 11/25/2013 Introduction to the theory of weak convergence Content. 1. σ-fields on metric spaces. 2. Kolmogorov σ-field on C[0;T ]. 3. Weak convergence. In the first two sections we review some concepts from measure theory on metric spaces. Then in the last section we begin the discussion of the theory of weak convergence, by stating and proving important Portmentau Theorem, which gives four equivalent definitions of weak convergence. 1 Borel σ-fields on metric space We consider a metric space (S; ρ). To discuss probability measures on metric spaces we first need to introduce σ-fields. Definition 1. Borel σ-field B on S is the field generated by the set of all open sets U ⊂ S. Lemma 1. Suppose S is Polish. Then every open set U ⊂ S can be represented as a countable union of balls B(x; r). Proof. Since S is Polish, we can identify a countable set x1; : : : ; xn;::: which is dense. For each xi 2 U, since U is open we can find a radius ri such that ∗ B(xi; ri) ⊂ U. Fix a constant M > 0. Consider ri = min(arg supfr : B(x; r) ⊂ Ug;M) > 0 and set r = r∗=2. Then [ B(x ; r ) ⊂ U. In i i i:xi2U i i order to finish the proof we need to show that equality [i:xi2U B(xi; ri) = U holds. Consider any x 2 U. There exists 0 < r ≤ M such that B(x; r) ⊂ U. -
Countable Product of Function Spaces Having P-Frechet-Urysohn Like Properties
Countable product of function spaces having p-Frechet-Urysohn like properties 著者 Tamariz-Mascarua Angel journal or Tsukuba journal of mathematics publication title volume 20 number 2 page range 291-319 year 1996-12 URL http://hdl.handle.net/2241/6940 TSUKUBA J.MATH. Vol. 20 No. 2 (1996), 291-319 COUNTABLE PRODUCT OF FUNCTION SPACES HAVING o-FRECHET-URYSOHN LIKE PROPERTIES By Angel Tamariz-Mascarua Abstract. We exhibit in this article some classes of spaces for which properties y and yp are countable additive, and we prove that for some type of spaces and ultrafiltersp e co*,y is equivalent to yp. We obtain: (1) If {Xn}n<(0 is a sequence of metrizable locally compact spaces with yp(pe(O*), then Yln<aCn(Xn) is a FU(p)- space; (2) Cn(X) is a Frechet-Urysohn (resp., FU(p)) space iff CK{F(X)) has the same property, where F(X) is the free topological group generated by X; (3) For a locally compact metrizable and non countable space X, CK(X) is a Frechet-Urysohn (resp., FU{p)) space iff Cn(LK(X)) is Frechet-Urysohn (resp., FU(p)), where LK(X) is the dual space of CK{X); (4) For every Cech-complete space X and every pecv* for which R does not have yp, CX{X) is Frechet-Urysohn iff CK{X) is a FU(p)-space. Also we give some results concerning F-points in (Q* related with /?-Frechet-Urysohn orooertv and toooloeical function soaces. 0. Introduction In [GN], [G], [Me], [Py], the authors studied the properties needed in a space X in order to have the Frechet-Urysohn property in the space Cn{X) of continuous functions from X to the real line R considered with the pointwise convergence topology. -
ON SEPARABLE SUPPORTS of BOREL MEASURES Let E
GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 1, 1995, 45-53 ON SEPARABLE SUPPORTS OF BOREL MEASURES A. KHARAZISHVILI Abstract. Some properties of Borel measures with separable sup- ports are considered. In particular, it is proved that any σ-finite Borel measure on a Suslin line has a separable supports and from this fact it is deduced, using the continuum hypothesis, that any Suslin line contains a Luzin subspace with the cardinality of the continuum. Let E be a topological space. We say that the space E has the pro- perty (S) if for every σ-finite Borel measure µ defined in this space there exists a separable support, i.e., a separable closed set F (µ) E such that µ(E F (µ)) = 0. ⊂ Letn us consider some examples of topological spaces having the property (S). Example 1. It is obvious that any separable topological space E has the property (S). Example 2. Let E be an arbitrary metric space whose topological weight is not measurable in a wide sense. Then according to the well-known result from the topological measure theory the space E has the property (S). Example 3. Let E be the Alexandrov compactification of some discrete topological space. Then the following statements are equivalent: a) the space E has the property (S); b) card(E) is not measurable in a wide sense. Example 4. Let E be a Hausdorff topological space. We say that E is a Luzin space if every σ-finite diffused (i.e., continuous) Borel measure defined in E is identically zero. The classical Luzin set on the real line R is a Luzin topological space (about Luzin sets see, for example, [1]). -
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. -
The Gap Between Gromov-Vague and Gromov-Hausdorff-Vague Topology
THE GAP BETWEEN GROMOV-VAGUE AND GROMOV-HAUSDORFF-VAGUE TOPOLOGY SIVA ATHREYA, WOLFGANG LOHR,¨ AND ANITA WINTER Abstract. In [ALW15] an invariance principle is stated for a class of strong Markov processes on tree-like metric measure spaces. It is shown that if the underlying spaces converge Gromov vaguely, then the processes converge in the sense of finite dimensional distributions. Further, if the underly- ing spaces converge Gromov-Hausdorff vaguely, then the processes converge weakly in path space. In this paper we systematically introduce and study the Gromov-vague and the Gromov-Hausdorff-vague topology on the space of equivalence classes of metric boundedly finite measure spaces. The latter topology is closely related to the Gromov-Hausdorff-Prohorov metric which is defined on different equivalence classes of metric measure spaces. We explain the necessity of these two topologies via several examples, and close the gap between them. That is, we show that convergence in Gromov- vague topology implies convergence in Gromov-Hausdorff-vague topology if and only if the so-called lower mass-bound property is satisfied. Further- more, we prove and disprove Polishness of several spaces of metric measure spaces in the topologies mentioned above (summarized in Figure 1). As an application, we consider the Galton-Watson tree with critical off- spring distribution of finite variance conditioned to not get extinct, and construct the so-called Kallenberg-Kesten tree as the weak limit in Gromov- Hausdorff-vague topology when the edge length are scaled down to go to zero. Contents 1. Introduction 1 2. The Gromov-vague topology 6 3. -
Cech-Completeness and Related Properties of The
CECH-COMPLETENESSˇ AND RELATED PROPERTIES OF THE GENERALIZED COMPACT-OPEN TOPOLOGY L'UBICA HOLA´ AND LASZL´ O´ ZSILINSZKY Abstract. The generalized compact-open topology τC on partial continuous functions with closed domains in X and values in Y is studied. If Y is a non- ˇ ˇ countably compact Cech-complete space with a Gδ-diagonal, then τC is Cech- complete, sieve complete and satisfies the p-space property of Arhangel'skiˇı, respectively, if and only if X is Lindel¨ofand locally compact. Lindel¨ofness, paracompactness and normality of τC is also investigated. New results are obtained on Cech-completeness,ˇ sieve completeness and the p-space property for the compact-open topology on the space of continuous functions with a general range Y . 1. Introduction The generalized compact-open topology τC on the space of partial continuous functions with closed domains was introduced by J. Back in [Ba] in connection with investigating utility functions emerging in mathematical economics. It proved to be a useful tool in studying convergence of dynamic programming models [Wh], [La], as well as in applications to the theory of differential equations [BC]. This new interest in τC complements the attention paid to spaces of partial maps in the past [Za], [Ku1], [Ku2], [AA], [AB], [BB], [Se], and more recently in [Fi], [St], [KS], [BCH1], [BCH2], [DN1], [DN2], [HZ1], [HZ2]. Various topological properties of τC have already been established, e.g. separa- tion axioms in [Ho1], complete metrizability in [Ho2] and other completeness type properties in [HZ1] and [NZ], respectively; also, in [DN1], [DN2] the authors study topological properties of spaces of partial maps in a more general setting.