Appendix Polish Spaces and Standard Borel Spaces
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Descriptive Set Theory of Complete Quasi-Metric Spaces
数理解析研究所講究録 第 1790 巻 2012 年 16-30 16 Descriptive set theory of complete quasi-metric spaces Matthew de Brecht National Institute of Information and Communications Technology Kyoto, Japan Abstract We give a summary of results from our investigation into extending classical descriptive set theory to the entire class of countably based $T_{0}$ -spaces. Polish spaces play a central role in the descriptive set theory of metrizable spaces, and we suggest that countably based completely quasi-metrizable spaces, which we refer to as quasi-Polish spaces, play the central role in the extended theory. The class of quasi-Polish spaces is general enough to include both Polish spaces and $\omega$ -continuous domains, which have many applications in theoretical computer science. We show that quasi-Polish spaces are a very natural generalization of Polish spaces in terms of their topological characterizations and their complete- ness properties. In particular, a metrizable space is quasi-Polish if and only if it is Polish, and many classical theorems concerning Polish spaces, such as the Hausdorff-Kuratowski theorem, generalize to all quasi-Polish spaces. 1. Introduction Descriptive set theory has proven to be an invaluable tool for the study of separable metrizable spaces, and the techniques and results have been applied to many fields such as functional analysis, topological group theory, and math- ematical logic. Separable completely metrizable spaces, called Polish spaces, play a central role in classical descriptive set theory. These spaces include the space of natural numbers with the discrete topology, the real numbers with the Euclidean topology, as well as the separable Hilbert and Banach spaces. -
Measure and Integration (Under Construction)
Measure and Integration (under construction) Gustav Holzegel∗ April 24, 2017 Abstract These notes accompany the lecture course ”Measure and Integration” at Imperial College London (Autumn 2016). They follow very closely the text “Real-Analysis” by Stein-Shakarchi, in fact most proofs are simple rephrasings of the proofs presented in the aforementioned book. Contents 1 Motivation 3 1.1 Quick review of the Riemann integral . 3 1.2 Drawbacks of the class R, motivation of the Lebesgue theory . 4 1.2.1 Limits of functions . 5 1.2.2 Lengthofcurves ....................... 5 1.2.3 TheFundamentalTheoremofCalculus . 5 1.3 Measures of sets in R ......................... 6 1.4 LiteratureandFurtherReading . 6 2 Measure Theory: Lebesgue Measure in Rd 7 2.1 Preliminaries and Notation . 7 2.2 VolumeofRectanglesandCubes . 7 2.3 Theexteriormeasure......................... 9 2.3.1 Examples ........................... 9 2.3.2 Propertiesoftheexteriormeasure . 10 2.4 Theclassof(Lebesgue)measurablesets . 12 2.4.1 The property of countable additivity . 14 2.4.2 Regularity properties of the Lebesgue measure . 15 2.4.3 Invariance properties of the Lebesgue measure . 16 2.4.4 σ-algebrasandBorelsets . 16 2.5 Constructionofanon-measurableset. 17 2.6 MeasurableFunctions ........................ 19 2.6.1 Some abstract preliminary remarks . 19 2.6.2 Definitions and equivalent formulations of measurability . 19 2.6.3 Properties 1: Behaviour under compositions . 21 2.6.4 Properties 2: Behaviour under limits . 21 2.6.5 Properties3: Behaviourof sums and products . 22 ∗Imperial College London, Department of Mathematics, South Kensington Campus, Lon- don SW7 2AZ, United Kingdom. 1 2.6.6 Thenotionof“almosteverywhere”. 22 2.7 Building blocks of integration theory . -
Topology and Descriptive Set Theory
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector TOPOLOGY AND ITS APPLICATIONS ELSEVIER Topology and its Applications 58 (1994) 195-222 Topology and descriptive set theory Alexander S. Kechris ’ Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA Received 28 March 1994 Abstract This paper consists essentially of the text of a series of four lectures given by the author in the Summer Conference on General Topology and Applications, Amsterdam, August 1994. Instead of attempting to give a general survey of the interrelationships between the two subjects mentioned in the title, which would be an enormous and hopeless task, we chose to illustrate them in a specific context, that of the study of Bore1 actions of Polish groups and Bore1 equivalence relations. This is a rapidly growing area of research of much current interest, which has interesting connections not only with topology and set theory (which are emphasized here), but also to ergodic theory, group representations, operator algebras and logic (particularly model theory and recursion theory). There are four parts, corresponding roughly to each one of the lectures. The first contains a brief review of some fundamental facts from descriptive set theory. In the second we discuss Polish groups, and in the third the basic theory of their Bore1 actions. The last part concentrates on Bore1 equivalence relations. The exposition is essentially self-contained, but proofs, when included at all, are often given in the barest outline. Keywords: Polish spaces; Bore1 sets; Analytic sets; Polish groups; Bore1 actions; Bore1 equivalence relations 1. -
Borel Structure in Groups and Their Dualso
BOREL STRUCTURE IN GROUPS AND THEIR DUALSO GEORGE W. MACKEY Introduction. In the past decade or so much work has been done toward extending the classical theory of finite dimensional representations of com- pact groups to a theory of (not necessarily finite dimensional) unitary repre- sentations of locally compact groups. Among the obstacles interfering with various aspects of this program is the lack of a suitable natural topology in the "dual object"; that is in the set of equivalence classes of irreducible representations. One can introduce natural topologies but none of them seem to have reasonable properties except in extremely special cases. When the group is abelian for example the dual object itself is a locally compact abelian group. This paper is based on the observation that for certain purposes one can dispense with a topology in the dual object in favor of a "weaker struc- ture" and that there is a wide class of groups for which this weaker structure has very regular properties. If .S is any topological space one defines a Borel (or Baire) subset of 5 to be a member of the smallest family of subsets of 5 which includes the open sets and is closed with respect to the formation of complements and countable unions. The structure defined in 5 by its Borel sets we may call the Borel structure of 5. It is weaker than the topological structure in the sense that any one-to-one transformation of S onto 5 which preserves the topological structure also preserves the Borel structure whereas the converse is usually false. -
Dirichlet Is Natural Vincent Danos, Ilias Garnier
Dirichlet is Natural Vincent Danos, Ilias Garnier To cite this version: Vincent Danos, Ilias Garnier. Dirichlet is Natural. MFPS 31 - Mathematical Foundations of Program- ming Semantics XXXI, Jun 2015, Nijmegen, Netherlands. pp.137-164, 10.1016/j.entcs.2015.12.010. hal-01256903 HAL Id: hal-01256903 https://hal.archives-ouvertes.fr/hal-01256903 Submitted on 15 Jan 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. MFPS 2015 Dirichlet is natural Vincent Danos1 D´epartement d'Informatique Ecole normale sup´erieure Paris, France Ilias Garnier2 School of Informatics University of Edinburgh Edinburgh, United Kingdom Abstract Giry and Lawvere's categorical treatment of probabilities, based on the probabilistic monad G, offer an elegant and hitherto unexploited treatment of higher-order probabilities. The goal of this paper is to follow this formulation to reconstruct a family of higher-order probabilities known as the Dirichlet process. This family is widely used in non-parametric Bayesian learning. Given a Polish space X, we build a family of higher-order probabilities in G(G(X)) indexed by M ∗(X) the set of non-zero finite measures over X. The construction relies on two ingredients. -
Polish Spaces and Baire Spaces
Polish spaces and Baire spaces Jordan Bell [email protected] Department of Mathematics, University of Toronto June 27, 2014 1 Introduction These notes consist of me working through those parts of the first chapter of Alexander S. Kechris, Classical Descriptive Set Theory, that I think are impor- tant in analysis. Denote by N the set of positive integers. I do not talk about universal spaces like the Cantor space 2N, the Baire space NN, and the Hilbert cube [0; 1]N, or \localization", or about Polish groups. If (X; τ) is a topological space, the Borel σ-algebra of X, denoted by BX , is the smallest σ-algebra of subsets of X that contains τ. BX contains τ, and is closed under complements and countable unions, and rather than talking merely about Borel sets (elements of the Borel σ-algebra), we can be more specific by talking about open sets, closed sets, and sets that are obtained by taking countable unions and complements. Definition 1. An Fσ set is a countable union of closed sets. A Gδ set is a complement of an Fσ set. Equivalently, it is a countable intersection of open sets. If (X; d) is a metric space, the topology induced by the metric d is the topology generated by the collection of open balls. If (X; τ) is a topological space, a metric d on the set X is said to be compatible with τ if τ is the topology induced by d.A metrizable space is a topological space whose topology is induced by some metric, and a completely metrizable space is a topological space whose topology is induced by some complete metric. -
A Framework to Study Commutation Problems Bulletin De La S
BULLETIN DE LA S. M. F. ALFONS VAN DAELE A framework to study commutation problems Bulletin de la S. M. F., tome 106 (1978), p. 289-309 <http://www.numdam.org/item?id=BSMF_1978__106__289_0> © Bulletin de la S. M. F., 1978, tous droits réservés. L’accès aux archives de la revue « Bulletin de la S. M. F. » (http: //smf.emath.fr/Publications/Bulletin/Presentation.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/ conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Bull. Soc. math. France, 106, 1978, p. 289-309. A FRAMEWORK TO STUDY COMMUTATION PROBLEMS BY ALFONS VAN DAELE [Kath. Univ. Leuven] RESUME. — Soient A et B deux algebres involutives d'operateurs sur un espace hil- bertien j^, telles que chacune d'elles soit contenue dans Ie commutant de Fautre. On enonce des conditions suffisantes sur A et B, en termes de certaines applications lineaires, ri : A ->• ^ et n' : B -> J^, pour que chacune de ces algebres engendre Ie commutant de Fautre. Cette structure generalise d'une certaine facon celle d'une algebre hilbertienne a gauche; elle permet de traiter Ie cas ou Falgebre de von Neumann et son commutant n'ont pas la meme grandeur. ABSTRACT. — If A and B are commuting *-algebras of operators on a Hilbert space ^, conditions on A and B are formulated in terms of linear maps T| : A -> ^ and n" : B -»• ^ to ensure that A and B generate each others commutants. -
Descriptive Set Theory
Descriptive Set Theory David Marker Fall 2002 Contents I Classical Descriptive Set Theory 2 1 Polish Spaces 2 2 Borel Sets 14 3 E®ective Descriptive Set Theory: The Arithmetic Hierarchy 27 4 Analytic Sets 34 5 Coanalytic Sets 43 6 Determinacy 54 7 Hyperarithmetic Sets 62 II Borel Equivalence Relations 73 1 8 ¦1-Equivalence Relations 73 9 Tame Borel Equivalence Relations 82 10 Countable Borel Equivalence Relations 87 11 Hyper¯nite Equivalence Relations 92 1 These are informal notes for a course in Descriptive Set Theory given at the University of Illinois at Chicago in Fall 2002. While I hope to give a fairly broad survey of the subject we will be concentrating on problems about group actions, particularly those motivated by Vaught's conjecture. Kechris' Classical Descriptive Set Theory is the main reference for these notes. Notation: If A is a set, A<! is the set of all ¯nite sequences from A. Suppose <! σ = (a0; : : : ; am) 2 A and b 2 A. Then σ b is the sequence (a0; : : : ; am; b). We let ; denote the empty sequence. If σ 2 A<!, then jσj is the length of σ. If f : N ! A, then fjn is the sequence (f(0); : : :b; f(n ¡ 1)). If X is any set, P(X), the power set of X is the set of all subsets X. If X is a metric space, x 2 X and ² > 0, then B²(x) = fy 2 X : d(x; y) < ²g is the open ball of radius ² around x. Part I Classical Descriptive Set Theory 1 Polish Spaces De¯nition 1.1 Let X be a topological space. -
Notes Which Was Later Published As Springer Lecture Notes No.128: Simplifi- Cation Was Not an Issue, but the Validity of Tomita’S Claim
STRUCTURE OF VON NEUMANN ALGEBRAS OF TYPE III Masamichi Takesaki Abstract. In this lecture, we will show that to every von Neumann algebra M there corresponds unquely a covariant system M, ⌧, R, ✓ on the real line R in such a way that n o f ✓ s M = M , ⌧ ✓s = e− ⌧, M0 M = C, ◦ \ where C is the center off M. In the case that M fis a factor, we have the following commutative square of groups which describes the relation of several important groups such as thef unitary group U(M) of M, the normalizer U(M) of M in M and the cohomology group of the flow of weights: C, R, ✓ : { } e f 1 1 1 ? ? @ ? 1 ? U?(C) ✓ B1( ,?U(C)) 1 −−−−−! yT −−−−−! y −−−−−! ✓ Ry −−−−−! ? ? @ ? 1 U(?M) U(?M) ✓ Z1( ,?U(C)) 1 −−−−−! y −−−−−! y −−−−−! ✓ Ry −−−−−! Ad Ade ? ? ˙ ? @✓ 1 1 Int?(M) Cnft?r(M) H ( ?, U(C)) 1 −−−−−! y −−−−−! y −−−−−! ✓ Ry −−−−−! ? ? ? ?1 ?1 ?1 y y y Contents Lecture 0. History of Structure Analyis of von Neumann Algebras of Type III. Lecture 1. Covariant System and Crossed Product. Lecture 2: Duality for the Crossed Product by Abelian Groups. Lecture 3: Second Cohomology of Locally Compact Abelian Group. Lecture 4. Arveson Spectrum of an Action of a Locally Compact Abelian Group G on a von Neumann algebra M. 1 2 VON NEUMANN ALGEBRAS OF TYPE III Lecture 5: Connes Spectrum. Lecture 6: Examples. Lecture 7: Crossed Product Construction of a Factor. Lecture 8: Structure of a Factor of Type III. Lecture 9: Hilbert Space Bundle. Lecture 10: The Non-Commutative Flow of Weights on a von Neumann algebra M, I. -
COUNTABLE BOREL EQUIVALENCE RELATIONS Introduction. These
COUNTABLE BOREL EQUIVALENCE RELATIONS SIMON THOMAS AND SCOTT SCHNEIDER Introduction. These notes are based upon a day-long lecture workshop presented by Simon Thomas at the University of Ohio at Athens on November 17, 2007. The workshop served as an intensive introduction to the emerging theory of countable Borel equivalence relations. These notes are an updated and slightly expanded version of an earlier draft which was compiled from the lecture slides by Scott Schneider. 1. First Session 1.1. Standard Borel Spaces and Borel Equivalence Relations. A topological space is said to be Polish if it admits a complete, separable metric. If B is a σ-algebra of subsets of a given set X, then the pair (X, B) is called a standard Borel space if there exists a Polish topology T on X that generates B as its Borel σ-algebra; in which case, we write B = B(T ). For example, each of the sets R, [0, 1], NN, and 2N = P(N) is Polish in its natural topology, and so may be viewed, equipped with its corresponding Borel structure, as a standard Borel space. The abstraction involved in passing from a topology to its associated Borel structure is analagous to that of passing from a metric to its induced topology. Just as distinct metrics on a space may induce the same topology, distinct topologies may very well generate the same Borel σ-algebra. In a standard Borel space, then, one “remembers” only the Borel sets, and forgets which of them were open; it is natural therefore to imagine that any of them might have been, and indeed this is the case: Theorem 1.1.1. -
4 from Random Borel Functions to Random Closed Sets
Tel Aviv University, 2012 Measurability and continuity 63 4 From random Borel functions to random closed sets 4a Random measurable maps . 63 4b Random continuous functions revisited . 65 4c Random semicontinuous functions . 66 4d Randomclosedsets ................. 70 Hints to exercises ....................... 75 Index ............................. 75 A random Borel set is not a random element of the Borel σ-algebra, but a random closed set is a random element of a standard Borel space of closed sets. 4a Random measurable maps Indicators of random Borel sets are a special case of random Borel functions. The supremum of a random Borel function is measurable due to Sect. 3. In Sect. 2c random functions were treated as random elements of some spaces of functions (see 2c10–2c13), much more restricted than all functions or all Borel functions. And indeed, the indicator of a random Borel set cannot be treated in this way (since the random Borel set cannot, recall Sect. 2d). Now we proceed in the spirit of 2d3. A probability space (Ω, F,P ) is assumed to be given, but rarely mentioned.1 4a1 Definition. A random Borel function is an equivalence class of maps Ω → RR that contains the map ω → x → f(ω, x) for some (F × B(R))-measurable f : Ω × R → R. As before, “equivalent” means “differ on a negligible set only”. As before, it is convenient to denote both objects by f; just denote f(ω) = f(ω, ·) = x → f(ω, x). You may easily give an equivalent definition in the spirit of 2d2. Compare it with 1d26–1d27 and 2c10–2c13. -
Stationary Probability Measures and Topological Realizations
ISRAEL JOURNAL OF MATHEMATICS ?? (2013), 1{14 DOI: 10.1007/s??????-??????-?????? STATIONARY PROBABILITY MEASURES AND TOPOLOGICAL REALIZATIONS BY Clinton T. Conley∗ 584 Malott Hall Cornell University Ithaca, NY 14853 USA e-mail: [email protected] URL: http://www.math.cornell.edu/People/Faculty/conley.html AND Alexander S. Kechris∗∗ Mathematics 253-37 Caltech Pasadena, CA 91125 USA e-mail: [email protected] URL: http://www.math.caltech.edu/people/kechris.html AND Benjamin D. Miller∗ Institut f¨urmathematische Logik und Grundlagenforschung Fachbereich Mathematik und Informatik Universit¨atM¨unster Einsteinstraße 62 48149 M¨unster Germany e-mail: [email protected] URL: http://wwwmath.uni-muenster.de/u/ben.miller 1 2 C.T. CONLEY, A.S. KECHRIS AND B.D. MILLER Isr. J. Math. ABSTRACT We establish the generic inexistence of stationary Borel probability mea- sures for aperiodic Borel actions of countable groups on Polish spaces. Using this, we show that every aperiodic continuous action of a countable group on a compact Polish space has an invariant Borel set on which it has no σ-compact realization. 1. Introduction It is well known that every standard Borel space is Borel isomorphic to a σ- compact Polish space (see [Kec95, Theorem 15.6]), and that every Borel action of a countable discrete group on a standard Borel space is Borel isomorphic to a continuous action on a Polish space (see [Kec95, Theorem 13.1]). Here we consider obstacles to simultaneously achieving both results. In x2, we give elementary examples of continuous actions of countable groups on Polish spaces which are not Borel isomorphic to continuous actions on σ- compact Hausdorff spaces.