SEPARABILITY AND COMPLETENESS FOR THE WASSERSTEIN DISTANCE
|
|
- Rolf Nichols
- 5 years ago
- Views:
Transcription
1 SEPARABILITY AND COMPLETENESS FOR THE WASSERSTEIN DISTANCE FRANÇOIS BOLLEY Abstract. In this note we prove in an elementary way that the Wasserstein distances, which play a basic role in optimal transportation issues, turn some spaces of probability measures into separable complete metric spaces. introduction Let (, d) be a separable complete metric space and P be the set of Borel probability measures on. Given p > 0, the R {+ }-valued map W p defined on P P by W p (µ, ν) = inf π W p (µ, ν) = inf π ( d(x, y) p dπ(x, y) ) 1 p if 1 p d(x, y) p dπ(x, y) if 0 < p < 1, where π runs over the set of probability measures on with marginals µ and ν, defines a metric on the subset P p of measures µ in P such that d(x 0, x) p dµ(x) be finite for some (and hence any) x 0 in : it is called the Wasserstein distance of order p (see [1], [4], [5] or [6] for instance). These distances are strongly linked to the theory of optimal transportation and have been widely used in various applications to partial differential equations, functional inequalities and probability theory. Some of them involve probability measures on infinite dimensional spaces such as the Wiener space of R d -valued continuous functions on the interval [0, T ] (as in [3] for instance) or some sets of probability measures on a phase space; this is a motivation to the general framework considered in this note, in which we shall prove: Theorem. If (, d) is a separable complete metric space and p a positive number, then the metric space (P p, W p ) is separable and complete. The completeness property has been proven in [4] by comparing the W p distances with the weaker Prohorov distance for which the property is known, and in [1] by means of a deep result by Kolmogorov. Here we shall give a more direct and elementary argument. Let us actually note that these kind of properties can be studied as in [2] within the following broader scope of weighted spaces of probability measures. Let (, τ) be a topological space and ω be a real-valued continuous function on, bounded by below by a positive constant, and let P ω denote the set of Borel probability measures µ on such that ω(x) dµ(x) be finite. We equip P ω with the natural weak topology defined by the set C bω of real-valued continuous functions f on such that ω 1 f be bounded on : this topology, which will be denoted w-c bω, is defined by the seminorms µ sup f i (x) dµ(x) i=1,...,n 1
2 2 FRANÇOIS BOLLEY for any finite family f 1,..., f n of functions in C bω. Then one can prove that if the topological space (, τ) is separable (resp. separable and metrizable, resp. separable, metrizable and topologically complete), then so is (P ω, w-c bω ). Conversely if (P ω, w-c bω ) is separable (resp. separable, metrizable and topologically complete), then so is (, τ) if (, τ) is a priori metrizable. As in the case when ω = 1, that is, without weight, where they are known, these properties can be proven either by building some explicit distances on the considered spaces of probability measures, or by abstract functional methods as in [2]. In the case when (, d) is a separable complete metric space and ω = 1 + d(x 0, ) p for p > 0, then the w-c bω topology on the set P ω = P p is metrized by the distance W p : in particular (P p, W p ) is separable and topologically complete. The following two sections are devoted to a direct proof of the above theorem, which in particular ensures that (P p, W p ) is complete. 1. Separability In this section we prove that the metric space (P p, W p ) is separable if (, d) is a separable complete metric space and p is a positive number. If (x n ) n is a sequence dense in (, d) we actually prove that the countable set of measures of the form b n δ xn, where N is an integer number, the b n s are nonnegative rational numbers with unit sum and δ x stands for the point mass at x, is dense in (P p, W p ). Let indeed µ be a given measure in P p and ε be a given positive number. 1. We first approach µ by a measure µ 1 = numbers with a n = 1. a n δ xn where the a n s are nonnegative real For this we note that is covered by the balls B(x n, ε max(1,1/p) ) with centers x n and radius ε max(1,1/p), and is partitioned by the sets B n = B(x n, ε max(1,1/p) )\ B(x k, ε max(1,1/p) ), k n 1 so that the a n = µ[ B n ] have unit sum. Moreover sending each point in B n onto x n for each n defines a transport map between µ and µ 1 = consequently B n x x n p dµ(x) W p (µ, µ 1 ) ε. 2. Then we approach µ 1 by a measure µ 2 = rational numbers with b n = 1. a n δ xn with cost a n ε p max(1,1/p) = ε max(p,1) ; b n δ xn where the b n s are nonnegative
3 SEPARABILITY AND COMPLETENESS FOR THE WASSERSTEIN DISTANCE 3 First of all µ 1 belongs to P p since it is at finite W p distance from the measure µ in P p. Hence a n x n x 1 p = W p (µ 1, δ x1 ) max(p,1) is finite since δ x1 also belongs to P p. In particular there exists an integer N such that n=n+1 a n x n x 1 p ε max(p,1). For each 2 n N we now let b n be a nonnegative rational number such that and such that ( N ) 1an 0 a n b n ε max(p,1) a j x j x 1 p, b 1 = a 1 + j=1 (a n b n ) + n=2 n=n+1 be rational: in particular the b n s have unit sum. Moreover one can transport µ 1 onto µ 2 = b n δ xn by keeping a b n mass at x n for each n N and sending the remaining a n b n mass from x n onto x 1, and sending the whole a n mass from x n onto x 1 for each n N + 1; the associated cost is (a n b n ) x n x 1 p + a n x n x 1 p 2 ε max(p,1), so that n=n+1 W p (µ 1, µ 2 ) 2 ε. 3. To sum up we have approached µ by a measure µ 2 in P p, of the expected form and which is at most 3 ε distant in W p metric. a n 2. Completeness In this section we prove that the metric space (P p, W p ) is complete if (, d) is a separable complete metric space and p is a positive number. Let indeed (µ n ) n be a Cauchy sequence in (P p, W p ). 1. For p 1 we first prove that (µ n ) n is uniformly tight by adapting a classical proof of Ulam lemma. Let ε be a given positive number. We note that (µ n ) n is Cauchy in (P 1, W 1 ) since W 1 W p. Hence there exists N such that W 1 (µ n, µ N ) ε 2 for any n N so that, for any n, there exists j N such that W 1 (µ n, µ j ) ε 2. (1)
4 4 FRANÇOIS BOLLEY The finite family (µ j ) j N is uniformly tight by Ulam lemma, so there exist a compact set K such that µ j (K) 1 ε for any j N, whence q points x 1,..., x q in such that for any j N, where U = q B(x k, ε). µ j (U) 1 ε (2) Then let φ be the 1 ( ε -Lipschitz function defined on by φ(x) = d(x, U) ) +. 1 Given ε j and n, if π is any joint measure on with marginals µ j and µ n, then φ(x) dµ j (x) φ(y) dµ n (y) = dπ(x, y) (φ(x) φ(y)) 1 d(x, y) dπ(x, y); ε hence φ(x) dµ j (x) φ(y) dµ n (y) 1 ε W 1(µ j, µ n ). On the other hand 1 U φ 1 U ε where U ε = {x; d(x, U) < ε}, so φ(x) dµ j (x) µ j (U) and φ(y) dµ n (y) µ n (U ε ). Consequently µ n (U ε ) µ j (U) 1 ε W 1(µ j, µ n ). (3) Thus, by (1), (2) and (3), for any ε > 0 we have found q points x 1,..., x q such that for any n since U ε q B(x k, 2 ε). ( q µ n \ B(x k, 2 ε) ) 2 ε Therefore, replacing ε by ε 2 m 1 where m is any integer, there exist q(m) points x m 1,..., x m q(m) in such that for any n. In particular the set is such that for any n. µ n ( \ S) q(m) ( µ n \ m=1 S = + B ( x m k, ε 2 m)) ε 2 m q(m) m=1 q(m) ( µ n \ B ( x m k, ε 2 m) B ( x m k, ε 2 m)) m=1 ε 2 m = ε
5 SEPARABILITY AND COMPLETENESS FOR THE WASSERSTEIN DISTANCE 5 On the other hand, for any ρ, and choosing m such that ε 2 m ρ, the set S can be covered by the q(m) balls B(x m k, ε 2 m ) with radius ε 2 m ρ: in other words it is totally bounded, so that its closure S is compact since is complete. To sum up, the set S is compact and satisfies µ n ( \ S) ε for any n: this means that the sequence (µ n ) n is indeed uniformly tight. 2. We deduce from step 1 that (µ n ) n converges in (P p, W p ) in the case when p 1. Indeed (µ n ) n is uniformly tight by step 1, so by Prohorov theorem there exists a subsequence (µ n ) n of (µ n ) n converging to a probability measure µ on for the narrow weak topology. The distance W p (µ, µ n ) actually tends to 0 as n goes to infinity. Let indeed π n m be a probability measure on with marginals µ n and µ m, optimal in the sense that d(x, y) p dπ n m (x, y) = W p(µ n, µ m ) p. The sequence (µ n ) n is uniformly tight, hence so is (π n m ) n for given m. Thus by Prohorov theorem again there exists a subsequence (π n m ) n of (π n m ) n converging to a probability measure π m on for the narrow weak topology. Then by semicontinuity d(x, y) p dπ m (x, y) lim inf d(x, y) p dπ n n m (x, y) = lim inf W p(µ n, µ m ) p. + n + (4) But on one hand π n m has marginals µ n and µ m, so at the limit (in n ) π m has marginals µ and µ m ; hence W p (µ, µ m ) p d(x, y) p dπ m (x, y) (5) for any m. On the other hand the sequence (µ n ) n is Cauchy for the distance W p, so for any ε > 0 and n, m large enough W p (µ n, µ m ) ε. (6) It finally follows from (4), (5) and (6) that W p (µ, µ m ) ε for m large enough, which means that µ belongs to P p and that W p (µ, µ n ) indeed tends to 0 as n goes to infinity. Finally W p (µ n, µ) tends to 0 as n goes to infinity since the whole sequence (µ n ) n is Cauchy in (P p, W p ). 3. We deduce from step 2 that (µ n ) n converges in (P p, W p ) in the case when 0 < p < 1. Indeed d p is a distance on which defines the same topology as d and (, d p ) is complete if so is (, d). Moreover P p (, d) = P 1 (, d p ) and W p (, d) = W 1 (, d p ) in obvious notation. Thus, given an exponent p ]0, 1[ and a metric d on, the results associated with the exponent p and the metric d stem from the results proven in step 2 for the exponent 1 and the metric d p.
6 6 FRANÇOIS BOLLEY References [1] L. Ambrosio, N. Gigli, and G. Savaré. Gradient flows in metric spaces and in the spaces of probability measures. Birkhäuser, Basel, [2] F. Bolley. Applications du transport optimal à des problèmes de limites de champ moyen. Thèse de doctorat, Ecole Normale Supérieure de Lyon. Available at [3] F. Bolley. Quantitative concentration inequalities on sample path space for mean field interaction. Preprint available at [4] S. T. Rachev. Probability metrics and the stability of stochastic models. John Wiley and Sons, Chichester, [5] S. T. Rachev and L. Rüschendorf. Mass transportation problems. Vol I and II. Springer, New York, [6] C. Villani. Topics in optimal transportation, volume 58 of Grad. Stud. Math. AMS, Providence, Ecole Normale Supérieure de Lyon, Umpa (UMR 5669), 46 allée d Italie, F Lyon cedex 7 Current address: Institut de Mathématiques, LSP (UMR C5583), Université Paul Sabatier, Route de Narbonne, F Toulouse cedex 9 address: bolley@cict.fr
Contractive metrics for scalar conservation laws
Contractive metrics for scalar conservation laws François Bolley 1, Yann Brenier 2, Grégoire Loeper 34 Abstract We consider nondecreasing entropy solutions to 1-d scalar conservation laws and show that
More informationLocal semiconvexity of Kantorovich potentials on non-compact manifolds
Local semiconvexity of Kantorovich potentials on non-compact manifolds Alessio Figalli, Nicola Gigli Abstract We prove that any Kantorovich potential for the cost function c = d / on a Riemannian manifold
More informationContents 1. Introduction 1 2. Main results 3 3. Proof of the main inequalities 7 4. Application to random dynamical systems 11 References 16
WEIGHTED CSISZÁR-KULLBACK-PINSKER INEQUALITIES AND APPLICATIONS TO TRANSPORTATION INEQUALITIES FRANÇOIS BOLLEY AND CÉDRIC VILLANI Abstract. We strengthen the usual Csiszár-Kullback-Pinsker inequality by
More informationMetric Spaces and Topology
Chapter 2 Metric Spaces and Topology From an engineering perspective, the most important way to construct a topology on a set is to define the topology in terms of a metric on the set. This approach underlies
More informationProblem Set 2: Solutions Math 201A: Fall 2016
Problem Set 2: s Math 201A: Fall 2016 Problem 1. (a) Prove that a closed subset of a complete metric space is complete. (b) Prove that a closed subset of a compact metric space is compact. (c) Prove that
More informationWeak KAM pairs and Monge-Kantorovich duality
Advanced Studies in Pure Mathematics 47-2, 2007 Asymptotic Analysis and Singularities pp. 397 420 Weak KAM pairs and Monge-Kantorovich duality Abstract. Patrick Bernard and Boris Buffoni The dynamics of
More informationIntroduction to Empirical Processes and Semiparametric Inference Lecture 08: Stochastic Convergence
Introduction to Empirical Processes and Semiparametric Inference Lecture 08: Stochastic Convergence Michael R. Kosorok, Ph.D. Professor and Chair of Biostatistics Professor of Statistics and Operations
More information3 (Due ). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?
MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due ). Show that the open disk x 2 + y 2 < 1 is a countable union of planar elementary sets. Show that the closed disk x 2 + y 2 1 is a countable
More informationFunctional Analysis. Franck Sueur Metric spaces Definitions Completeness Compactness Separability...
Functional Analysis Franck Sueur 2018-2019 Contents 1 Metric spaces 1 1.1 Definitions........................................ 1 1.2 Completeness...................................... 3 1.3 Compactness......................................
More informationSUMMARY OF RESULTS ON PATH SPACES AND CONVERGENCE IN DISTRIBUTION FOR STOCHASTIC PROCESSES
SUMMARY OF RESULTS ON PATH SPACES AND CONVERGENCE IN DISTRIBUTION FOR STOCHASTIC PROCESSES RUTH J. WILLIAMS October 2, 2017 Department of Mathematics, University of California, San Diego, 9500 Gilman Drive,
More informationMATHS 730 FC Lecture Notes March 5, Introduction
1 INTRODUCTION MATHS 730 FC Lecture Notes March 5, 2014 1 Introduction Definition. If A, B are sets and there exists a bijection A B, they have the same cardinality, which we write as A, #A. If there exists
More informationOn a Class of Multidimensional Optimal Transportation Problems
Journal of Convex Analysis Volume 10 (2003), No. 2, 517 529 On a Class of Multidimensional Optimal Transportation Problems G. Carlier Université Bordeaux 1, MAB, UMR CNRS 5466, France and Université Bordeaux
More informationTHEOREMS, ETC., FOR MATH 515
THEOREMS, ETC., FOR MATH 515 Proposition 1 (=comment on page 17). If A is an algebra, then any finite union or finite intersection of sets in A is also in A. Proposition 2 (=Proposition 1.1). For every
More informationSTAT 7032 Probability Spring Wlodek Bryc
STAT 7032 Probability Spring 2018 Wlodek Bryc Created: Friday, Jan 2, 2014 Revised for Spring 2018 Printed: January 9, 2018 File: Grad-Prob-2018.TEX Department of Mathematical Sciences, University of Cincinnati,
More informationWiener Measure and Brownian Motion
Chapter 16 Wiener Measure and Brownian Motion Diffusion of particles is a product of their apparently random motion. The density u(t, x) of diffusing particles satisfies the diffusion equation (16.1) u
More informationREPRESENTABLE BANACH SPACES AND UNIFORMLY GÂTEAUX-SMOOTH NORMS. Julien Frontisi
Serdica Math. J. 22 (1996), 33-38 REPRESENTABLE BANACH SPACES AND UNIFORMLY GÂTEAUX-SMOOTH NORMS Julien Frontisi Communicated by G. Godefroy Abstract. It is proved that a representable non-separable Banach
More information(1) Consider the space S consisting of all continuous real-valued functions on the closed interval [0, 1]. For f, g S, define
Homework, Real Analysis I, Fall, 2010. (1) Consider the space S consisting of all continuous real-valued functions on the closed interval [0, 1]. For f, g S, define ρ(f, g) = 1 0 f(x) g(x) dx. Show that
More informationCHAPTER V DUAL SPACES
CHAPTER V DUAL SPACES DEFINITION Let (X, T ) be a (real) locally convex topological vector space. By the dual space X, or (X, T ), of X we mean the set of all continuous linear functionals on X. By the
More informationEmpirical Processes: General Weak Convergence Theory
Empirical Processes: General Weak Convergence Theory Moulinath Banerjee May 18, 2010 1 Extended Weak Convergence The lack of measurability of the empirical process with respect to the sigma-field generated
More informationA note on some approximation theorems in measure theory
A note on some approximation theorems in measure theory S. Kesavan and M. T. Nair Department of Mathematics, Indian Institute of Technology, Madras, Chennai - 600 06 email: kesh@iitm.ac.in and mtnair@iitm.ac.in
More informationAN ELEMENTARY PROOF OF THE TRIANGLE INEQUALITY FOR THE WASSERSTEIN METRIC
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 136, Number 1, January 2008, Pages 333 339 S 0002-9939(07)09020- Article electronically published on September 27, 2007 AN ELEMENTARY PROOF OF THE
More informationChapter 2 Metric Spaces
Chapter 2 Metric Spaces The purpose of this chapter is to present a summary of some basic properties of metric and topological spaces that play an important role in the main body of the book. 2.1 Metrics
More informationInvariant measures for iterated function systems
ANNALES POLONICI MATHEMATICI LXXV.1(2000) Invariant measures for iterated function systems by Tomasz Szarek (Katowice and Rzeszów) Abstract. A new criterion for the existence of an invariant distribution
More informationarxiv: v3 [stat.co] 22 Apr 2016
A fixed-point approach to barycenters in Wasserstein space Pedro C. Álvarez-Esteban1, E. del Barrio 1, J.A. Cuesta-Albertos 2 and C. Matrán 1 1 Departamento de Estadística e Investigación Operativa and
More informationE.7 Alaoglu s Theorem
E.7 Alaoglu s Theorem 359 E.7 Alaoglu s Theorem If X is a normed space, then the closed unit ball in X or X is compact if and only if X is finite-dimensional (Problem A.25). Even so, Alaoglu s Theorem
More informationREGULARITY OF MONOTONE TRANSPORT MAPS BETWEEN UNBOUNDED DOMAINS
REGULARITY OF MONOTONE TRANSPORT MAPS BETWEEN UNBOUNDED DOMAINS DARIO CORDERO-ERAUSQUIN AND ALESSIO FIGALLI A Luis A. Caffarelli en su 70 años, con amistad y admiración Abstract. The regularity of monotone
More informationRecall that if X is a compact metric space, C(X), the space of continuous (real-valued) functions on X, is a Banach space with the norm
Chapter 13 Radon Measures Recall that if X is a compact metric space, C(X), the space of continuous (real-valued) functions on X, is a Banach space with the norm (13.1) f = sup x X f(x). We want to identify
More information2 (Bonus). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?
MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due 9/5). Prove that every countable set A is measurable and µ(a) = 0. 2 (Bonus). Let A consist of points (x, y) such that either x or y is
More informationWeak convergence and Brownian Motion. (telegram style notes) P.J.C. Spreij
Weak convergence and Brownian Motion (telegram style notes) P.J.C. Spreij this version: December 8, 2006 1 The space C[0, ) In this section we summarize some facts concerning the space C[0, ) of real
More informationWeak convergence and Compactness.
Chapter 4 Weak convergence and Compactness. Let be a complete separable metic space and B its Borel σ field. We denote by M() the space of probability measures on (, B). A sequence µ n M() of probability
More informationIntroduction and Preliminaries
Chapter 1 Introduction and Preliminaries This chapter serves two purposes. The first purpose is to prepare the readers for the more systematic development in later chapters of methods of real analysis
More informationOPTIMAL TRANSPORTATION PLANS AND CONVERGENCE IN DISTRIBUTION
OPTIMAL TRANSPORTATION PLANS AND CONVERGENCE IN DISTRIBUTION J.A. Cuesta-Albertos 1, C. Matrán 2 and A. Tuero-Díaz 1 1 Departamento de Matemáticas, Estadística y Computación. Universidad de Cantabria.
More information1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3
Index Page 1 Topology 2 1.1 Definition of a topology 2 1.2 Basis (Base) of a topology 2 1.3 The subspace topology & the product topology on X Y 3 1.4 Basic topology concepts: limit points, closed sets,
More informationAn introduction to some aspects of functional analysis
An introduction to some aspects of functional analysis Stephen Semmes Rice University Abstract These informal notes deal with some very basic objects in functional analysis, including norms and seminorms
More informationTools from Lebesgue integration
Tools from Lebesgue integration E.P. van den Ban Fall 2005 Introduction In these notes we describe some of the basic tools from the theory of Lebesgue integration. Definitions and results will be given
More informationOn the distributional divergence of vector fields vanishing at infinity
Proceedings of the Royal Society of Edinburgh, 141A, 65 76, 2011 On the distributional divergence of vector fields vanishing at infinity Thierry De Pauw Institut de Recherches en Mathématiques et Physique,
More informationProhorov s theorem. Bengt Ringnér. October 26, 2008
Prohorov s theorem Bengt Ringnér October 26, 2008 1 The theorem Definition 1 A set Π of probability measures defined on the Borel sets of a topological space is called tight if, for each ε > 0, there is
More informationMaths 212: Homework Solutions
Maths 212: Homework Solutions 1. The definition of A ensures that x π for all x A, so π is an upper bound of A. To show it is the least upper bound, suppose x < π and consider two cases. If x < 1, then
More informationContinuity of convex functions in normed spaces
Continuity of convex functions in normed spaces In this chapter, we consider continuity properties of real-valued convex functions defined on open convex sets in normed spaces. Recall that every infinitedimensional
More informationSome SDEs with distributional drift Part I : General calculus. Flandoli, Franco; Russo, Francesco; Wolf, Jochen
Title Author(s) Some SDEs with distributional drift Part I : General calculus Flandoli, Franco; Russo, Francesco; Wolf, Jochen Citation Osaka Journal of Mathematics. 4() P.493-P.54 Issue Date 3-6 Text
More informationMAT 578 FUNCTIONAL ANALYSIS EXERCISES
MAT 578 FUNCTIONAL ANALYSIS EXERCISES JOHN QUIGG Exercise 1. Prove that if A is bounded in a topological vector space, then for every neighborhood V of 0 there exists c > 0 such that tv A for all t > c.
More informationINTRODUCTION TO TOPOLOGY, MATH 141, PRACTICE PROBLEMS
INTRODUCTION TO TOPOLOGY, MATH 141, PRACTICE PROBLEMS Problem 1. Give an example of a non-metrizable topological space. Explain. Problem 2. Introduce a topology on N by declaring that open sets are, N,
More information(convex combination!). Use convexity of f and multiply by the common denominator to get. Interchanging the role of x and y, we obtain that f is ( 2M ε
1. Continuity of convex functions in normed spaces In this chapter, we consider continuity properties of real-valued convex functions defined on open convex sets in normed spaces. Recall that every infinitedimensional
More informationThe optimal partial transport problem
The optimal partial transport problem Alessio Figalli Abstract Given two densities f and g, we consider the problem of transporting a fraction m [0, min{ f L 1, g L 1}] of the mass of f onto g minimizing
More informationParcours OJD, Ecole Polytechnique et Université Pierre et Marie Curie 05 Mai 2015
Examen du cours Optimisation Stochastique Version 06/05/2014 Mastère de Mathématiques de la Modélisation F. Bonnans Parcours OJD, Ecole Polytechnique et Université Pierre et Marie Curie 05 Mai 2015 Authorized
More informationREAL AND COMPLEX ANALYSIS
REAL AND COMPLE ANALYSIS Third Edition Walter Rudin Professor of Mathematics University of Wisconsin, Madison Version 1.1 No rights reserved. Any part of this work can be reproduced or transmitted in any
More informationMath 426 Homework 4 Due 3 November 2017
Math 46 Homework 4 Due 3 November 017 1. Given a metric space X,d) and two subsets A,B, we define the distance between them, dista,b), as the infimum inf a A, b B da,b). a) Prove that if A is compact and
More informationStone-Čech compactification of Tychonoff spaces
The Stone-Čech compactification of Tychonoff spaces Jordan Bell jordan.bell@gmail.com Department of Mathematics, University of Toronto June 27, 2014 1 Completely regular spaces and Tychonoff spaces A topological
More informationA note on the σ-algebra of cylinder sets and all that
A note on the σ-algebra of cylinder sets and all that José Luis Silva CCM, Univ. da Madeira, P-9000 Funchal Madeira BiBoS, Univ. of Bielefeld, Germany (luis@dragoeiro.uma.pt) September 1999 Abstract In
More informationFrom now on, we will represent a metric space with (X, d). Here are some examples: i=1 (x i y i ) p ) 1 p, p 1.
Chapter 1 Metric spaces 1.1 Metric and convergence We will begin with some basic concepts. Definition 1.1. (Metric space) Metric space is a set X, with a metric satisfying: 1. d(x, y) 0, d(x, y) = 0 x
More informationTopological vectorspaces
(July 25, 2011) Topological vectorspaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ Natural non-fréchet spaces Topological vector spaces Quotients and linear maps More topological
More informationis a Borel subset of S Θ for each c R (Bertsekas and Shreve, 1978, Proposition 7.36) This always holds in practical applications.
Stat 811 Lecture Notes The Wald Consistency Theorem Charles J. Geyer April 9, 01 1 Analyticity Assumptions Let { f θ : θ Θ } be a family of subprobability densities 1 with respect to a measure µ on a measurable
More informationNECESSARY CONDITIONS FOR WEIGHTED POINTWISE HARDY INEQUALITIES
NECESSARY CONDITIONS FOR WEIGHTED POINTWISE HARDY INEQUALITIES JUHA LEHRBÄCK Abstract. We establish necessary conditions for domains Ω R n which admit the pointwise (p, β)-hardy inequality u(x) Cd Ω(x)
More informationP-adic Functions - Part 1
P-adic Functions - Part 1 Nicolae Ciocan 22.11.2011 1 Locally constant functions Motivation: Another big difference between p-adic analysis and real analysis is the existence of nontrivial locally constant
More informationSome Background Material
Chapter 1 Some Background Material In the first chapter, we present a quick review of elementary - but important - material as a way of dipping our toes in the water. This chapter also introduces important
More informationThe Arzelà-Ascoli Theorem
John Nachbar Washington University March 27, 2016 The Arzelà-Ascoli Theorem The Arzelà-Ascoli Theorem gives sufficient conditions for compactness in certain function spaces. Among other things, it helps
More informationA description of transport cost for signed measures
A description of transport cost for signed measures Edoardo Mainini Abstract In this paper we develop the analysis of [AMS] about the extension of the optimal transport framework to the space of real measures.
More informationCompact operators on Banach spaces
Compact operators on Banach spaces Jordan Bell jordan.bell@gmail.com Department of Mathematics, University of Toronto November 12, 2017 1 Introduction In this note I prove several things about compact
More informationTYPICAL BOREL MEASURES ON [0, 1] d SATISFY A MULTIFRACTAL FORMALISM
TYPICAL BOREL MEASURES ON [0, 1] d SATISFY A MULTIFRACTAL FORMALISM Abstract. In this article, we prove that in the Baire category sense, measures supported by the unit cube of R d typically satisfy a
More informationMeasure and Integration: Solutions of CW2
Measure and Integration: s of CW2 Fall 206 [G. Holzegel] December 9, 206 Problem of Sheet 5 a) Left (f n ) and (g n ) be sequences of integrable functions with f n (x) f (x) and g n (x) g (x) for almost
More informationAnother Riesz Representation Theorem
Another Riesz Representation Theorem In these notes we prove (one version of) a theorem known as the Riesz Representation Theorem. Some people also call it the Riesz Markov Theorem. It expresses positive
More information4 Countability axioms
4 COUNTABILITY AXIOMS 4 Countability axioms Definition 4.1. Let X be a topological space X is said to be first countable if for any x X, there is a countable basis for the neighborhoods of x. X is said
More informationStat 8112 Lecture Notes Weak Convergence in Metric Spaces Charles J. Geyer January 23, Metric Spaces
Stat 8112 Lecture Notes Weak Convergence in Metric Spaces Charles J. Geyer January 23, 2013 1 Metric Spaces Let X be an arbitrary set. A function d : X X R is called a metric if it satisfies the folloing
More informationECARES Université Libre de Bruxelles MATH CAMP Basic Topology
ECARES Université Libre de Bruxelles MATH CAMP 03 Basic Topology Marjorie Gassner Contents: - Subsets, Cartesian products, de Morgan laws - Ordered sets, bounds, supremum, infimum - Functions, image, preimage,
More informationIt follows from the above inequalities that for c C 1
3 Spaces L p 1. In this part we fix a measure space (, A, µ) (we may assume that µ is complete), and consider the A -measurable functions on it. 2. For f L 1 (, µ) set f 1 = f L 1 = f L 1 (,µ) = f dµ.
More informationReal Analysis Notes. Thomas Goller
Real Analysis Notes Thomas Goller September 4, 2011 Contents 1 Abstract Measure Spaces 2 1.1 Basic Definitions........................... 2 1.2 Measurable Functions........................ 2 1.3 Integration..............................
More informationLECTURE 15: COMPLETENESS AND CONVEXITY
LECTURE 15: COMPLETENESS AND CONVEXITY 1. The Hopf-Rinow Theorem Recall that a Riemannian manifold (M, g) is called geodesically complete if the maximal defining interval of any geodesic is R. On the other
More informationContents: 1. Minimization. 2. The theorem of Lions-Stampacchia for variational inequalities. 3. Γ -Convergence. 4. Duality mapping.
Minimization Contents: 1. Minimization. 2. The theorem of Lions-Stampacchia for variational inequalities. 3. Γ -Convergence. 4. Duality mapping. 1 Minimization A Topological Result. Let S be a topological
More informationStationary states for the aggregation equation with power law attractive-repulsive potentials
Stationary states for the aggregation equation with power law attractive-repulsive potentials Daniel Balagué Joint work with J.A. Carrillo, T. Laurent and G. Raoul Universitat Autònoma de Barcelona BIRS
More informationAnalysis Finite and Infinite Sets The Real Numbers The Cantor Set
Analysis Finite and Infinite Sets Definition. An initial segment is {n N n n 0 }. Definition. A finite set can be put into one-to-one correspondence with an initial segment. The empty set is also considered
More informationg 2 (x) (1/3)M 1 = (1/3)(2/3)M.
COMPACTNESS If C R n is closed and bounded, then by B-W it is sequentially compact: any sequence of points in C has a subsequence converging to a point in C Conversely, any sequentially compact C R n is
More informationWeak convergence. Amsterdam, 13 November Leiden University. Limit theorems. Shota Gugushvili. Generalities. Criteria
Weak Leiden University Amsterdam, 13 November 2013 Outline 1 2 3 4 5 6 7 Definition Definition Let µ, µ 1, µ 2,... be probability measures on (R, B). It is said that µ n converges weakly to µ, and we then
More informationMath 209B Homework 2
Math 29B Homework 2 Edward Burkard Note: All vector spaces are over the field F = R or C 4.6. Two Compactness Theorems. 4. Point Set Topology Exercise 6 The product of countably many sequentally compact
More informationPCA sets and convexity
F U N D A M E N T A MATHEMATICAE 163 (2000) PCA sets and convexity by Robert K a u f m a n (Urbana, IL) Abstract. Three sets occurring in functional analysis are shown to be of class PCA (also called Σ
More informationHomework Assignment #2 for Prob-Stats, Fall 2018 Due date: Monday, October 22, 2018
Homework Assignment #2 for Prob-Stats, Fall 2018 Due date: Monday, October 22, 2018 Topics: consistent estimators; sub-σ-fields and partial observations; Doob s theorem about sub-σ-field measurability;
More informationREVIEW OF ESSENTIAL MATH 346 TOPICS
REVIEW OF ESSENTIAL MATH 346 TOPICS 1. AXIOMATIC STRUCTURE OF R Doğan Çömez The real number system is a complete ordered field, i.e., it is a set R which is endowed with addition and multiplication operations
More informationOptimal transportation on non-compact manifolds
Optimal transportation on non-compact manifolds Albert Fathi, Alessio Figalli 07 November 2007 Abstract In this work, we show how to obtain for non-compact manifolds the results that have already been
More informationSpaces with Ricci curvature bounded from below
Spaces with Ricci curvature bounded from below Nicola Gigli February 23, 2015 Topics 1) On the definition of spaces with Ricci curvature bounded from below 2) Analytic properties of RCD(K, N) spaces 3)
More informationUniqueness of the solution to the Vlasov-Poisson system with bounded density
Uniqueness of the solution to the Vlasov-Poisson system with bounded density Grégoire Loeper December 16, 2005 Abstract In this note, we show uniqueness of weak solutions to the Vlasov- Poisson system
More informationMixed exterior Laplace s problem
Mixed exterior Laplace s problem Chérif Amrouche, Florian Bonzom Laboratoire de mathématiques appliquées, CNRS UMR 5142, Université de Pau et des Pays de l Adour, IPRA, Avenue de l Université, 64000 Pau
More informationCLASS NOTES FOR APRIL 14, 2000
CLASS NOTES FOR APRIL 14, 2000 Announcement: Section 1.2, Questions 3,5 have been deferred from Assignment 1 to Assignment 2. Section 1.4, Question 5 has been dropped entirely. 1. Review of Wednesday class
More informationApplied Analysis (APPM 5440): Final exam 1:30pm 4:00pm, Dec. 14, Closed books.
Applied Analysis APPM 44: Final exam 1:3pm 4:pm, Dec. 14, 29. Closed books. Problem 1: 2p Set I = [, 1]. Prove that there is a continuous function u on I such that 1 ux 1 x sin ut 2 dt = cosx, x I. Define
More informationIt follows from the above inequalities that for c C 1
3 Spaces L p 1. Appearance of normed spaces. In this part we fix a measure space (, A, µ) (we may assume that µ is complete), and consider the A - measurable functions on it. 2. For f L 1 (, µ) set f 1
More informationANSWER TO A QUESTION BY BURR AND ERDŐS ON RESTRICTED ADDITION, AND RELATED RESULTS Mathematics Subject Classification: 11B05, 11B13, 11P99
ANSWER TO A QUESTION BY BURR AND ERDŐS ON RESTRICTED ADDITION, AND RELATED RESULTS N. HEGYVÁRI, F. HENNECART AND A. PLAGNE Abstract. We study the gaps in the sequence of sums of h pairwise distinct elements
More informationMeasurable Choice Functions
(January 19, 2013) Measurable Choice Functions Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/fun/choice functions.pdf] This note
More informationb i (µ, x, s) ei ϕ(x) µ s (dx) ds (2) i=1
NONLINEAR EVOLTION EQATIONS FOR MEASRES ON INFINITE DIMENSIONAL SPACES V.I. Bogachev 1, G. Da Prato 2, M. Röckner 3, S.V. Shaposhnikov 1 The goal of this work is to prove the existence of a solution to
More informationDynkin (λ-) and π-systems; monotone classes of sets, and of functions with some examples of application (mainly of a probabilistic flavor)
Dynkin (λ-) and π-systems; monotone classes of sets, and of functions with some examples of application (mainly of a probabilistic flavor) Matija Vidmar February 7, 2018 1 Dynkin and π-systems Some basic
More informationLecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University
Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University February 7, 2007 2 Contents 1 Metric Spaces 1 1.1 Basic definitions...........................
More informationAnalysis III Theorems, Propositions & Lemmas... Oh My!
Analysis III Theorems, Propositions & Lemmas... Oh My! Rob Gibson October 25, 2010 Proposition 1. If x = (x 1, x 2,...), y = (y 1, y 2,...), then is a distance. ( d(x, y) = x k y k p Proposition 2. In
More informationMath 201 Topology I. Lecture notes of Prof. Hicham Gebran
Math 201 Topology I Lecture notes of Prof. Hicham Gebran hicham.gebran@yahoo.com Lebanese University, Fanar, Fall 2015-2016 http://fs2.ul.edu.lb/math http://hichamgebran.wordpress.com 2 Introduction and
More informationFinal. due May 8, 2012
Final due May 8, 2012 Write your solutions clearly in complete sentences. All notation used must be properly introduced. Your arguments besides being correct should be also complete. Pay close attention
More informationGLUING LEMMAS AND SKOROHOD REPRESENTATIONS
GLUING LEMMAS AND SKOROHOD REPRESENTATIONS PATRIZIA BERTI, LUCA PRATELLI, AND PIETRO RIGO Abstract. Let X, E), Y, F) and Z, G) be measurable spaces. Suppose we are given two probability measures γ and
More informationYour first day at work MATH 806 (Fall 2015)
Your first day at work MATH 806 (Fall 2015) 1. Let X be a set (with no particular algebraic structure). A function d : X X R is called a metric on X (and then X is called a metric space) when d satisfies
More informationCHAPTER I THE RIESZ REPRESENTATION THEOREM
CHAPTER I THE RIESZ REPRESENTATION THEOREM We begin our study by identifying certain special kinds of linear functionals on certain special vector spaces of functions. We describe these linear functionals
More informationPolishness of Weak Topologies Generated by Gap and Excess Functionals
Journal of Convex Analysis Volume 3 (996), No. 2, 283 294 Polishness of Weak Topologies Generated by Gap and Excess Functionals Ľubica Holá Mathematical Institute, Slovak Academy of Sciences, Štefánikovà
More informationSolutions to Tutorial 8 (Week 9)
The University of Sydney School of Mathematics and Statistics Solutions to Tutorial 8 (Week 9) MATH3961: Metric Spaces (Advanced) Semester 1, 2018 Web Page: http://www.maths.usyd.edu.au/u/ug/sm/math3961/
More informationWEAK CURVATURE CONDITIONS AND FUNCTIONAL INEQUALITIES
WEA CURVATURE CODITIOS AD FUCTIOAL IEQUALITIES JOH LOTT AD CÉDRIC VILLAI Abstract. We give sufficient conditions for a measured length space (, d, ν to admit local and global Poincaré inequalities, along
More informationOptimal Transport for Data Analysis
Optimal Transport for Data Analysis Bernhard Schmitzer 2017-05-16 1 Introduction 1.1 Reminders on Measure Theory Reference: Ambrosio, Fusco, Pallara: Functions of Bounded Variation and Free Discontinuity
More informationBERNARD HOST AND BRYNA KRA
UIFORMITY SEMIORMS O l AD A IVERSE THEOREM SUMMARY OF RESULTS BERARD HOST AD BRYA KRA Abstract. For each integer k, we define seminorms on l (Z), analogous to the seminorms defined by the authors on bounded
More informationA generic property of families of Lagrangian systems
Annals of Mathematics, 167 (2008), 1099 1108 A generic property of families of Lagrangian systems By Patrick Bernard and Gonzalo Contreras * Abstract We prove that a generic Lagrangian has finitely many
More information