The Locale of Random Sequences
|
|
- Daniela White
- 5 years ago
- Views:
Transcription
1 1/31 The Locale of Random Sequences Alex Simpson LFCS, School of Informatics University of Edinburgh, UK
2 The Locale of Random Sequences 2/31 The Application Problem in probability What is the empirical meaning of event E having probability p? Standard explanation, cf. Kolmogorov (1933): Perform large number of repeated trials. Proportion of occurrences of event E is practically certain to be close to p. Practical certainty means with probability very close to 1. (Of course, can calculate exact probability that observed value lies within ɛ of p.) What is the empirical meaning of practical certainty??? A vicious circle!
3 The Locale of Random Sequences 3/31 Random sequences Explaining the empirical meaning of probability amounts to explaining the characteristic properties of sequences of stochastic events. So need a theory of random sequences. Intuitively, a sequence α 2 ω (where 2 := {0, 1}) is random if it could potentially arise by tossing a fair coin (we use 0 for heads and 1 for tails) ad infinitum. We seek a mathematical theory of random sequences R 2 ω. Task is also interesting for more general sequences of stochastic events. E.g., tossing a sequence of biased coins.
4 The Locale of Random Sequences 4/31 Questions such a theory should answer Is the sequence 0 ω random? Is the bit-sequence of Chaitin s halting probability Ω random? Under what conditions does a transducer on infinite sequences preserve randomness? Suppose we toss a sequence of biased coins, where the i-th coin has probability δ i of coming up 0 and 1 2 δ i of coming up 1. Under what conditions on (δ i ) are we guaranteed to obtain a fair random sequence? (van Lambalgen s brainteaser, 1987) Such questions seem meaningless in probability theory à la Kolmogorov
5 The Locale of Random Sequences 5/31 von Mises theory of probability Richard von Mises (1919) developed a mathematical theory of probability, based directly on random sequences. He postulated that random sequences should be characterised by principles encapsulating two contrasting aspects of a random α 2 ω : Local irregularity: e.g., cannot predict α n from α 0... α n 1 Global regularity: frequencies converge lim n 1 n n 1 i=0 α i = 1 2 (this embodies the law of large numbers) In von Mises theory probability is defined as the limit of observed frequencies.
6 The Locale of Random Sequences 6/31 Difficulties von Mises attempt at formalizing random sequences (whence probability) met with mathematical difficulties. Local irregularity is formalized as invariance under admissible place selections. Formulating this precisely is not straightforward. The approach is limited in power; e.g., cannot derive the full law of the iterated logarithm from von Mises principles. The theory is a mathematical outcast. Care is needed to avoid inconsistency. It does not sit easily within set theory. There are also philosophical issues. For example, the law of large numbers is an idealistic rather than empirical assumption, since we can never observe that it holds.
7 The Locale of Random Sequences 7/31 Observable properties We seek a theory of random sequences based only on our empirical experience of sequences of stochastic events. At any time, we only see a finite prefix of a random sequence (and this might be any finite sequence of digits). A subset U 2 ω is open (in the Cantor/product topology) if it satisfies: α U = n. β 2 ω. (β n = α n = β U) Open subsets correspond to observable properties, cf. Smyth, Abramsky. For example, {α α 0 ω } is observable. But {0 ω } is not observable.
8 The Locale of Random Sequences 8/31 Empirical properties of randomness An empirical fact: If α is a random sequence, then we will sooner or later observe that α 0 ω, i.e., that α {β 2 ω β 0 ω }. Note that, for any n, the probability that α n = 0 n is 2 n. Thus the probability that the prefix α n fails to show that α 0 ω is 2 n. We soon have to have been very unlucky not to have observed that α 0 ω. The empirical fact is explained by a general principle that the observer is not infinitely unlucky. We turn this principle into a (first) postulate of randomness.
9 The Locale of Random Sequences 9/31 Postulates of randomness We postulate properties of the desired set R 2 ω of random sequences. Let λ be the uniform/lebesgue measure on 2 ω. (R1) First postulate of randomness For every open U 2 ω, if λ(u) = 1 then R U. Informally: if α is a random sequence, and U is an almost sure observable property then we will (eventually) observe that α U. The first postulate addresses the Application Problem. The theory predicts that any almost sure observation will actually occur. Thus almost sure becomes sure for open sets.
10 The Locale of Random Sequences 10/31 The second postulate is a nontriviality condition. Its purpose is to ensure that there are enough random sequences. (R2) Second postulate of randomness For every open U 2 ω, if R U then λ(u) = 1. Informally: if U is an observable property with λ(u) < 1 then there has to be a random sequence outside U since there is some finite probability of randomly generating such a sequence. (There is some similarity with Myhill s set-theoretic axioms for randomness, cf. van Lambalgen (1992). Myhill s axioms, however, use measurable sets rather than open sets and invoke set-theoretic definability. I believe it is easier to justify the above postulates as embodying empirical properties of randomness.)
11 The Locale of Random Sequences 11/31 Inconsistency The two postulates of randomness are inconsistent. For any sequence β, the open set {α α β} has measure 1. Therefore, by the first postulate of randomness: R {α α β}, i.e., β R. Thus R =, but this contradicts the second postulate.
12 The Locale of Random Sequences 12/31 Conventional wisdom Ideally, a random sequence would be one satisfying all measure 1 properties (irrespective of whether open or not). There are no such sequences. It is therefore necessary to restrict to a a countable family F of (not necessarily open) measure 1 subsets of 2 ω. Define: R F := F, Because F is countable, we have λ(r F ) = 1, hence nontriviality. N.B., our first postulate of randomness is weakened, but the second holds as stated.
13 The Locale of Random Sequences 13/31 Algorithmic and logical notions of randomness Typically the family F is given using either recursion theoretic or logical notions of definability. Numerous such definitions have been proposed and argued for; for example, by Martin-Löf (1966), Schnorr (1971), Kurtz (1981),... Algorithmic randomness is an important area of recursion theory. Does it, however, provide the right framework for modelling the stochastic phenomenon of randomness? Two criticisms: No one canonical notion of randomness While one can make a (dubious!) case that recursion-theoretic restrictions reflect our observational limitations, there is no reason to believe any recursion-theoretic dependencies to be inherent in the stochastic phenomenon of randomness itself
14 The Locale of Random Sequences 14/31 Aim of talk The goal of the talk is to present a mathematically natural approach to modelling the stochastic phenomenon of randomness. A notable feature is that our two postulates of randomness will hold as originally formulated. In particular, there is no need to select a countable family F of measure 1 sets. Conventional wisdom is wrong
15 The Locale of Random Sequences 15/31 Some clues The two postulates of randomness are couched in terms of observable properties (open sets). Thus it is natural to define randomness in a setting in which observable properties play a fundamental role. Further, we never experience a completed infinite random sequence in its entirety only its finite prefixes. Thus it is natural to define randomness in a setting in which completed entities (points) are not the basic ingredient. All this suggests using locale theory (a.k.a. point-free topology)
16 The Locale of Random Sequences 16/31 Locales A locale X is given by a frame O(X), that is, O(X) is a partially-ordered set with: arbitrary joins (including the empty join ), (hence) finite meets (including the empty meet X), satisfying the distributive law: U ( i I V i ) = i I U V i. Motivating example: O(X) is the lattice of opens of a topological space, joins are unions and finite meets are intersections.
17 The Locale of Random Sequences 17/31 Locales as generalised spaces Locale theory abstracts away from topological spaces. The frame O(X) does not need to arise as a family of sets under union and intersection. Indeed there need not be any underlying set of points. The generality can be motivated by considering frames as theories in a natural logic of observable properties, see Vickers (1989). Many notions from topology transfer to locales, X, Y. A (continuous) map f : X Y is given by a function f 1 : O(Y ) O(X) that preserves arbitrary joins and finite meets. A map f : X Y is said to be an embedding if the function f 1 : O(Y ) O(X) is surjective. The embeddings determine the notion of sublocale.
18 The Locale of Random Sequences 18/31 The two postulates interpreted for locales Recall the two postulates. (R1) For every open U 2 ω, if λ(u) = 1 then R U. (R2) For every open U 2 ω, if R U then λ(u) = 1. 2 ω is a topological space, hence a locale. The open subsets U 2 ω are the elements of O(2 ω ), they determine the open sublocales of 2 ω. By interpreting as the sublocale relationship, the postulates make sense verbatim for locales, but the inconsistency vanishes. Theorem and (R2). There exists a unique sublocale R 2 ω satisfying (R1)
19 The Locale of Random Sequences 19/31 Concrete description of R For U, V 2 ω open, define: U V : λ(u) = λ(u V ) = λ(v ) If U V then intuitively they represent the same observation on random sequences. Define O(R) := O(2 ω )/. (Cf. the measure (σ-)algebra of 2 ω, equivalently of [0, 1].) Proposition The above defines R as a sublocale of 2 ω. This is the sublocale characterised by the previous theorem.
20 The Locale of Random Sequences 20/31 Outer measure We postulated that random sequences satisfy all measure 1 observable properties. What about other measure 1 properties? The outer measure of a sublocale Y 2 ω is defined by: λ (Y ) := inf{λ(u) U O(2 ω ) and Y U} (R2) asserts that R has outer measure 1. Proposition R is characterised as the smallest sublocale of 2 ω of outer measure 1. In particular, random sequences satisfy all measure 1 properties. In contrast, in algorithmic randomness, the assumption that a sequence satisfies all measure 1 effective open properties (i.e., that it is Kurtz random) does not even imply the law of large numbers.
21 The Locale of Random Sequences 21/31 What are the random sequences in R? The locale R has no points, that is there are no maps from the one point (i.e., terminal) locale to R. Thus, according to this theory, there is no such thing as a completed infinite random sequence. In particular, 0 ω is not random. Also, the bit-sequence of Chaitin s Ω is not random. Nonethless R is a nontrivial space of random sequences. (More generally, is the only compact sublocale of R.)
22 The Locale of Random Sequences 22/31 What are the maps from R to R? Intuitively, these correspond to continuous transducers on random sequences. Theorem The maps R R are in one-to-one correspondence with the continuous nonsingular maps from 2 ω to 2 ω modulo almost everywhere equivalence. A continuous nonsingular map from 2 ω to 2 ω is a continuous function from a measure 1 subspace D 2 ω to 2 ω, satisfying: for every null Z 2 ω, it holds that f 1 (Z) is null.
23 The Locale of Random Sequences 23/31 More generally If we generalise λ to other probability measures (technically, probability valuations), and 2 ω to other locales, we still obtain canonical random sublocales. A (continuous) probability valuation on a locale X is a (necessarily monotone) function µ: O(X) [0, 1] satisfying: µ( ) = 0 µ(u V ) = µ(u) + µ(v ) µ(u V ) µ(x) = 1 µ( U) = sup µ(u) (D O(X) directed) Given (X, µ) define: U D U D U µ V : µ(u V ) = µ(u V ) O(R(µ)) := O(X)/ µ
24 The Locale of Random Sequences 24/31 Proposition R(µ) is a sublocale of X. Define the outer value of a sublocale Y X by: µ (Y ) := inf{µ(u) U O(X) and Y U} Theorem If the locale X is regular then: 1. R(µ) is the meet of all open sublocales U X with µ(u) = 1 2. R(µ) is the meet of all sublocales Y X with µ (Y ) = 1 3. µ (R(µ)) = 1 Obviously, µ is also a probability valuation on R(µ). Then (R(µ), µ) is random in the sense that: [U] [V ] O(R(µ)) (proper inclusion) µ([u]) < µ([v ])
25 The Locale of Random Sequences 25/31 When does one measure subsume another? For 2 ω, valuations are in one-one correspondence with Borel measures. Proposition Given two probability measures µ, ν on 2 ω, the following are equivalent: R(µ) R(ν) (as sublocales of 2 ω ) µ ν (i.e., µ is absolutely continuous relative to ν) Recall, µ is absolutely continuous relative to ν if every ν-null Borel set is also µ-null.
26 The Locale of Random Sequences 26/31 van Lambalgen s brainteaser revisited Suppose we toss a sequence of biased coins, where the i-th coin has probability δ i of coming up 0 and 1 2 δ i of coming up 1. Under what conditions on (δ i ) are we guaranteed to obtain a fair random sequence? We write π for the product measure i ( δ i, 1 2 δ i) on 2 ω. The question asks: When does it hold that R(π) R? Solution: R(π) R π λ δ 2 i < i=0 (by previous proposition) (by Kakutani s Theorem)
27 The Locale of Random Sequences 27/31 Another characterisation of R The locale of random sequences, R, is: countably based zero dimensional and has no points Moreover, the valuation λ on R is random. Canonicity Theorem If X is a countably-based zero-dimensional locale with no points and X carries a random probability valuation µ then X and R are homeomorphic via valuation preserving maps. (Cf. the measure algebra isomorphism theorem of measure theory.)
28 The Locale of Random Sequences 28/31 Constructive version? Should be possible to develop the theory within Bishop Constructive Mathematics, and hence in any brand of constructivism. In contrast, the usual theories of algorithmic randomness are not compatible with (formal) Church s Thesis. In a constructive version, an existence proof for a map R R should give rise to an effective transducer of random sequences. Given a non-atomic effective probability valuation µ on 2 ω, a constructive proof of the canonicity theorem should give mutually inverse transducers R(µ) R and R R(µ); the former amounting to an optimal data compression, and the latter its decoding. Thus a constructive canonicity theorem should relate to Shannon s noiseless coding theorem from information theory.
29 The Locale of Random Sequences 29/31 Random space? If space S is continuous, how does this account for the existence of discrete observables S {0, 1}? If space is discrete, how does this account for its geometry, e.g., straight lines, geodesics, etc.? Perhaps space is random... A motivating idea for this is that any attempt to zoom in indefinitely on the location of a particle must eventually, at some level of granularity, settle down as a sequence of stochastic events.
30 The Locale of Random Sequences 30/31 Random space! Let R(E n ) be the smallest thick sublocale of E n (Euclidean n-space). Equivalently R(E n ) = n {}}{ R(E)#... #R(E) where # is the independent product of random locales. The space R(E n ) is zero dimensional, so have many nontrivial discrete observables R(E n ) {0, 1}. R(E n ) also possesses a nontrivial geometry. For example, it has random straight lines through independent points: (x, y, λ) λx + (1 λ)y : R(E n )#R(E n )#R(E) R(E n )
31 The Locale of Random Sequences 31/31 Other directions Ideally, the development presented here should (once constructivised) be part of a general constructive account of probability and measure theory over locales. Aspects of such a theory have been initiated by, amongst others, Coquand, Palmgren, Spitters and Vickers. Much remains to be done. In recent talks, Dana Scott has identified the frame O(R) as a subframe of the measure algebra, and used this to give a sheaf model of a (classical) modal set theory (MZF) in which every proposition has a probability. This opens the possibility of performing modal set-theoretic reasoning about randomness.
Part V. 17 Introduction: What are measures and why measurable sets. Lebesgue Integration Theory
Part V 7 Introduction: What are measures and why measurable sets Lebesgue Integration Theory Definition 7. (Preliminary). A measure on a set is a function :2 [ ] such that. () = 2. If { } = is a finite
More informationChapter 2 Classical Probability Theories
Chapter 2 Classical Probability Theories In principle, those who are not interested in mathematical foundations of probability theory might jump directly to Part II. One should just know that, besides
More informationConnectedness. Proposition 2.2. The following are equivalent for a topological space (X, T ).
Connectedness 1 Motivation Connectedness is the sort of topological property that students love. Its definition is intuitive and easy to understand, and it is a powerful tool in proofs of well-known results.
More informationModule 1. Probability
Module 1 Probability 1. Introduction In our daily life we come across many processes whose nature cannot be predicted in advance. Such processes are referred to as random processes. The only way to derive
More informationThe Logical Approach to Randomness
The Logical Approach to Randomness Christopher P. Porter University of Florida UC Irvine C-ALPHA Seminar May 12, 2015 Introduction The concept of randomness plays an important role in mathematical practice,
More informationMATH41011/MATH61011: FOURIER SERIES AND LEBESGUE INTEGRATION. Extra Reading Material for Level 4 and Level 6
MATH41011/MATH61011: FOURIER SERIES AND LEBESGUE INTEGRATION Extra Reading Material for Level 4 and Level 6 Part A: Construction of Lebesgue Measure The first part the extra material consists of the construction
More informationLecture 11: Random Variables
EE5110: Probability Foundations for Electrical Engineers July-November 2015 Lecture 11: Random Variables Lecturer: Dr. Krishna Jagannathan Scribe: Sudharsan, Gopal, Arjun B, Debayani The study of random
More informationMeasure and Integration: Concepts, Examples and Exercises. INDER K. RANA Indian Institute of Technology Bombay India
Measure and Integration: Concepts, Examples and Exercises INDER K. RANA Indian Institute of Technology Bombay India Department of Mathematics, Indian Institute of Technology, Bombay, Powai, Mumbai 400076,
More informationMetric spaces and metrizability
1 Motivation Metric spaces and metrizability By this point in the course, this section should not need much in the way of motivation. From the very beginning, we have talked about R n usual and how relatively
More informationThe Lebesgue Integral
The Lebesgue Integral Brent Nelson In these notes we give an introduction to the Lebesgue integral, assuming only a knowledge of metric spaces and the iemann integral. For more details see [1, Chapters
More informationInvariance Properties of Random Sequences
Invariance Properties of Random Sequences Peter Hertling (Department of Computer Science, The University of Auckland, Private Bag 92019, Auckland, New Zealand Email: hertling@cs.auckland.ac.nz) Yongge
More informationPROBABILITY THEORY 1. Basics
PROILITY THEORY. asics Probability theory deals with the study of random phenomena, which under repeated experiments yield different outcomes that have certain underlying patterns about them. The notion
More informationTT-FUNCTIONALS AND MARTIN-LÖF RANDOMNESS FOR BERNOULLI MEASURES
TT-FUNCTIONALS AND MARTIN-LÖF RANDOMNESS FOR BERNOULLI MEASURES LOGAN AXON Abstract. For r [0, 1], the Bernoulli measure µ r on the Cantor space {0, 1} N assigns measure r to the set of sequences with
More informationThe small ball property in Banach spaces (quantitative results)
The small ball property in Banach spaces (quantitative results) Ehrhard Behrends Abstract A metric space (M, d) is said to have the small ball property (sbp) if for every ε 0 > 0 there exists a sequence
More informationThe equivalence axiom and univalent models of type theory.
The equivalence axiom and univalent models of type theory. (Talk at CMU on February 4, 2010) By Vladimir Voevodsky Abstract I will show how to define, in any type system with dependent sums, products and
More informationII - REAL ANALYSIS. This property gives us a way to extend the notion of content to finite unions of rectangles: we define
1 Measures 1.1 Jordan content in R N II - REAL ANALYSIS Let I be an interval in R. Then its 1-content is defined as c 1 (I) := b a if I is bounded with endpoints a, b. If I is unbounded, we define c 1
More informationRS Chapter 1 Random Variables 6/5/2017. Chapter 1. Probability Theory: Introduction
Chapter 1 Probability Theory: Introduction Basic Probability General In a probability space (Ω, Σ, P), the set Ω is the set of all possible outcomes of a probability experiment. Mathematically, Ω is just
More information18.175: Lecture 2 Extension theorems, random variables, distributions
18.175: Lecture 2 Extension theorems, random variables, distributions Scott Sheffield MIT Outline Extension theorems Characterizing measures on R d Random variables Outline Extension theorems Characterizing
More informationA PECULIAR COIN-TOSSING MODEL
A PECULIAR COIN-TOSSING MODEL EDWARD J. GREEN 1. Coin tossing according to de Finetti A coin is drawn at random from a finite set of coins. Each coin generates an i.i.d. sequence of outcomes (heads or
More informationSequence convergence, the weak T-axioms, and first countability
Sequence convergence, the weak T-axioms, and first countability 1 Motivation Up to now we have been mentioning the notion of sequence convergence without actually defining it. So in this section we will
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 informationUniversity of Sheffield. School of Mathematics & and Statistics. Measure and Probability MAS350/451/6352
University of Sheffield School of Mathematics & and Statistics Measure and Probability MAS350/451/6352 Spring 2018 Chapter 1 Measure Spaces and Measure 1.1 What is Measure? Measure theory is the abstract
More informationContinuum Probability and Sets of Measure Zero
Chapter 3 Continuum Probability and Sets of Measure Zero In this chapter, we provide a motivation for using measure theory as a foundation for probability. It uses the example of random coin tossing to
More informationProbability. Lecture Notes. Adolfo J. Rumbos
Probability Lecture Notes Adolfo J. Rumbos October 20, 204 2 Contents Introduction 5. An example from statistical inference................ 5 2 Probability Spaces 9 2. Sample Spaces and σ fields.....................
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 informationChapter 1. Measure Spaces. 1.1 Algebras and σ algebras of sets Notation and preliminaries
Chapter 1 Measure Spaces 1.1 Algebras and σ algebras of sets 1.1.1 Notation and preliminaries We shall denote by X a nonempty set, by P(X) the set of all parts (i.e., subsets) of X, and by the empty set.
More informationLecture 20 : Markov Chains
CSCI 3560 Probability and Computing Instructor: Bogdan Chlebus Lecture 0 : Markov Chains We consider stochastic processes. A process represents a system that evolves through incremental changes called
More informationMeasure and integration
Chapter 5 Measure and integration In calculus you have learned how to calculate the size of different kinds of sets: the length of a curve, the area of a region or a surface, the volume or mass of a solid.
More informationChapter One. The Real Number System
Chapter One. The Real Number System We shall give a quick introduction to the real number system. It is imperative that we know how the set of real numbers behaves in the way that its completeness and
More informationPart III. 10 Topological Space Basics. Topological Spaces
Part III 10 Topological Space Basics Topological Spaces Using the metric space results above as motivation we will axiomatize the notion of being an open set to more general settings. Definition 10.1.
More informationMath 676. A compactness theorem for the idele group. and by the product formula it lies in the kernel (A K )1 of the continuous idelic norm
Math 676. A compactness theorem for the idele group 1. Introduction Let K be a global field, so K is naturally a discrete subgroup of the idele group A K and by the product formula it lies in the kernel
More informationSemantics of intuitionistic propositional logic
Semantics of intuitionistic propositional logic Erik Palmgren Department of Mathematics, Uppsala University Lecture Notes for Applied Logic, Fall 2009 1 Introduction Intuitionistic logic is a weakening
More informationProbabilistic Observations and Valuations (Extended Abstract) 1
Replace this ile with prentcsmacro.sty or your meeting, or with entcsmacro.sty or your meeting. Both can be ound at the ENTCS Macro Home Page. Probabilistic Observations and Valuations (Extended Abstract)
More informationUNIFORM TEST OF ALGORITHMIC RANDOMNESS OVER A GENERAL SPACE
UNIFORM TEST OF ALGORITHMIC RANDOMNESS OVER A GENERAL SPACE PETER GÁCS ABSTRACT. The algorithmic theory of randomness is well developed when the underlying space is the set of finite or infinite sequences
More informationMeasure Theoretic Probability. P.J.C. Spreij
Measure Theoretic Probability P.J.C. Spreij this version: September 16, 2009 Contents 1 σ-algebras and measures 1 1.1 σ-algebras............................... 1 1.2 Measures...............................
More information2.2 Annihilators, complemented subspaces
34CHAPTER 2. WEAK TOPOLOGIES, REFLEXIVITY, ADJOINT OPERATORS 2.2 Annihilators, complemented subspaces Definition 2.2.1. [Annihilators, Pre-Annihilators] Assume X is a Banach space. Let M X and N X. We
More informationThe Metamathematics of Randomness
The Metamathematics of Randomness Jan Reimann January 26, 2007 (Original) Motivation Effective extraction of randomness In my PhD-thesis I studied the computational power of reals effectively random for
More informationI. ANALYSIS; PROBABILITY
ma414l1.tex Lecture 1. 12.1.2012 I. NLYSIS; PROBBILITY 1. Lebesgue Measure and Integral We recall Lebesgue measure (M411 Probability and Measure) λ: defined on intervals (a, b] by λ((a, b]) := b a (so
More information6 Classical dualities and reflexivity
6 Classical dualities and reflexivity 1. Classical dualities. Let (Ω, A, µ) be a measure space. We will describe the duals for the Banach spaces L p (Ω). First, notice that any f L p, 1 p, generates the
More informationTopology Proceedings. COPYRIGHT c by Topology Proceedings. All rights reserved.
Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: topolog@auburn.edu ISSN: 0146-4124
More informationA Non-Topological View of Dcpos as Convergence Spaces
A Non-Topological View of Dcpos as Convergence Spaces Reinhold Heckmann AbsInt Angewandte Informatik GmbH, Stuhlsatzenhausweg 69, D-66123 Saarbrücken, Germany e-mail: heckmann@absint.com Abstract The category
More informationUnivalent Foundations and Set Theory
Univalent Foundations and Set Theory Talk by Vladimir Voevodsky from Institute for Advanced Study in Princeton, NJ. May 8, 2013 1 Univalent foundations - are based on a class of formal deduction systems
More informationOverview of normed linear spaces
20 Chapter 2 Overview of normed linear spaces Starting from this chapter, we begin examining linear spaces with at least one extra structure (topology or geometry). We assume linearity; this is a natural
More informationTree sets. Reinhard Diestel
1 Tree sets Reinhard Diestel Abstract We study an abstract notion of tree structure which generalizes treedecompositions of graphs and matroids. Unlike tree-decompositions, which are too closely linked
More information1.1. MEASURES AND INTEGRALS
CHAPTER 1: MEASURE THEORY In this chapter we define the notion of measure µ on a space, construct integrals on this space, and establish their basic properties under limits. The measure µ(e) will be defined
More informationLogics above S4 and the Lebesgue measure algebra
Logics above S4 and the Lebesgue measure algebra Tamar Lando Abstract We study the measure semantics for propositional modal logics, in which formulas are interpreted in the Lebesgue measure algebra M,
More informationMeasures. Chapter Some prerequisites. 1.2 Introduction
Lecture notes Course Analysis for PhD students Uppsala University, Spring 2018 Rostyslav Kozhan Chapter 1 Measures 1.1 Some prerequisites I will follow closely the textbook Real analysis: Modern Techniques
More informationOn minimal models of the Region Connection Calculus
Fundamenta Informaticae 69 (2006) 1 20 1 IOS Press On minimal models of the Region Connection Calculus Lirong Xia State Key Laboratory of Intelligent Technology and Systems Department of Computer Science
More informationPOL502: Foundations. Kosuke Imai Department of Politics, Princeton University. October 10, 2005
POL502: Foundations Kosuke Imai Department of Politics, Princeton University October 10, 2005 Our first task is to develop the foundations that are necessary for the materials covered in this course. 1
More informationMeasures. 1 Introduction. These preliminary lecture notes are partly based on textbooks by Athreya and Lahiri, Capinski and Kopp, and Folland.
Measures These preliminary lecture notes are partly based on textbooks by Athreya and Lahiri, Capinski and Kopp, and Folland. 1 Introduction Our motivation for studying measure theory is to lay a foundation
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 informationLecture 1: Overview of percolation and foundational results from probability theory 30th July, 2nd August and 6th August 2007
CSL866: Percolation and Random Graphs IIT Delhi Arzad Kherani Scribe: Amitabha Bagchi Lecture 1: Overview of percolation and foundational results from probability theory 30th July, 2nd August and 6th August
More informationMath 541 Fall 2008 Connectivity Transition from Math 453/503 to Math 541 Ross E. Staffeldt-August 2008
Math 541 Fall 2008 Connectivity Transition from Math 453/503 to Math 541 Ross E. Staffeldt-August 2008 Closed sets We have been operating at a fundamental level at which a topological space is a set together
More information5 Measure theory II. (or. lim. Prove the proposition. 5. For fixed F A and φ M define the restriction of φ on F by writing.
5 Measure theory II 1. Charges (signed measures). Let (Ω, A) be a σ -algebra. A map φ: A R is called a charge, (or signed measure or σ -additive set function) if φ = φ(a j ) (5.1) A j for any disjoint
More informationLinear Algebra. Paul Yiu. Department of Mathematics Florida Atlantic University. Fall 2011
Linear Algebra Paul Yiu Department of Mathematics Florida Atlantic University Fall 2011 Linear Algebra Paul Yiu Department of Mathematics Florida Atlantic University Fall 2011 1A: Vector spaces Fields
More informationChapter 4. Measure Theory. 1. Measure Spaces
Chapter 4. Measure Theory 1. Measure Spaces Let X be a nonempty set. A collection S of subsets of X is said to be an algebra on X if S has the following properties: 1. X S; 2. if A S, then A c S; 3. if
More informationRandomness Beyond Lebesgue Measure
Randomness Beyond Lebesgue Measure Jan Reimann Department of Mathematics University of California, Berkeley November 16, 2006 Measures on Cantor Space Outer measures from premeasures Approximate sets from
More informationthe time it takes until a radioactive substance undergoes a decay
1 Probabilities 1.1 Experiments with randomness Wewillusethetermexperimentinaverygeneralwaytorefertosomeprocess that produces a random outcome. Examples: (Ask class for some first) Here are some discrete
More informationSynthetic Probability Theory
Synthetic Probability Theory Alex Simpson Faculty of Mathematics and Physics University of Ljubljana, Slovenia PSSL, Amsterdam, October 2018 Gian-Carlo Rota (1932-1999): The beginning definitions in any
More informationStat 451: Solutions to Assignment #1
Stat 451: Solutions to Assignment #1 2.1) By definition, 2 Ω is the set of all subsets of Ω. Therefore, to show that 2 Ω is a σ-algebra we must show that the conditions of the definition σ-algebra are
More informationStatistical Inference
Statistical Inference Lecture 1: Probability Theory MING GAO DASE @ ECNU (for course related communications) mgao@dase.ecnu.edu.cn Sep. 11, 2018 Outline Introduction Set Theory Basics of Probability Theory
More informationBinary positivity in the language of locales
Binary positivity in the language of locales Francesco Ciraulo Department of Mathematics University of Padua 4 th Workshop on Formal Topology June 15-20 2012, Ljubljana Francesco Ciraulo (Padua) Binary
More informationTopology and experimental distinguishability
Topology and experimental distinguishability Gabriele Carcassi Christine A Aidala David J Baker Mark J Greenfield University of Michigan Motivation for this work This work has its root in an effort to
More informationConstruction of a general measure structure
Chapter 4 Construction of a general measure structure We turn to the development of general measure theory. The ingredients are a set describing the universe of points, a class of measurable subsets along
More informationDiscrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 14
CS 70 Discrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 14 Introduction One of the key properties of coin flips is independence: if you flip a fair coin ten times and get ten
More informationLebesgue Measure on R n
CHAPTER 2 Lebesgue Measure on R n Our goal is to construct a notion of the volume, or Lebesgue measure, of rather general subsets of R n that reduces to the usual volume of elementary geometrical sets
More informationVIC 2004 p.1/26. Institut für Informatik Universität Heidelberg. Wolfgang Merkle and Jan Reimann. Borel Normality, Automata, and Complexity
Borel Normality, Automata, and Complexity Wolfgang Merkle and Jan Reimann Institut für Informatik Universität Heidelberg VIC 2004 p.1/26 The Quest for Randomness Intuition: An infinite sequence of fair
More informationLecture 10: Probability distributions TUESDAY, FEBRUARY 19, 2019
Lecture 10: Probability distributions DANIEL WELLER TUESDAY, FEBRUARY 19, 2019 Agenda What is probability? (again) Describing probabilities (distributions) Understanding probabilities (expectation) Partial
More informationA bitopological point-free approach to compactifications
A bitopological point-free approach to compactifications Olaf Karl Klinke a, Achim Jung a, M. Andrew Moshier b a School of Computer Science University of Birmingham Birmingham, B15 2TT England b School
More informationRandom Reals à la Chaitin with or without prefix-freeness
Random Reals à la Chaitin with or without prefix-freeness Verónica Becher Departamento de Computación, FCEyN Universidad de Buenos Aires - CONICET Argentina vbecher@dc.uba.ar Serge Grigorieff LIAFA, Université
More informationMuchnik Degrees: Results and Techniques
Muchnik Degrees: Results and Techniques Stephen G. Simpson Department of Mathematics Pennsylvania State University http://www.math.psu.edu/simpson/ Draft: April 1, 2003 Abstract Let P and Q be sets of
More informationRVs and their probability distributions
RVs and their probability distributions RVs and their probability distributions In these notes, I will use the following notation: The probability distribution (function) on a sample space will be denoted
More information2. Prime and Maximal Ideals
18 Andreas Gathmann 2. Prime and Maximal Ideals There are two special kinds of ideals that are of particular importance, both algebraically and geometrically: the so-called prime and maximal ideals. Let
More informationA NEW SET THEORY FOR ANALYSIS
Article A NEW SET THEORY FOR ANALYSIS Juan Pablo Ramírez 0000-0002-4912-2952 Abstract: We present the real number system as a generalization of the natural numbers. First, we prove the co-finite topology,
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 informationFormal Epistemology: Lecture Notes. Horacio Arló-Costa Carnegie Mellon University
Formal Epistemology: Lecture Notes Horacio Arló-Costa Carnegie Mellon University hcosta@andrew.cmu.edu Logical preliminaries Let L 0 be a language containing a complete set of Boolean connectives, including
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 3 9/10/2008 CONDITIONING AND INDEPENDENCE
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 3 9/10/2008 CONDITIONING AND INDEPENDENCE Most of the material in this lecture is covered in [Bertsekas & Tsitsiklis] Sections 1.3-1.5
More informationCHAPTER 7. Connectedness
CHAPTER 7 Connectedness 7.1. Connected topological spaces Definition 7.1. A topological space (X, T X ) is said to be connected if there is no continuous surjection f : X {0, 1} where the two point set
More informationHilbert spaces. 1. Cauchy-Schwarz-Bunyakowsky inequality
(October 29, 2016) Hilbert spaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/fun/notes 2016-17/03 hsp.pdf] Hilbert spaces are
More informationLebesgue Measure on R n
8 CHAPTER 2 Lebesgue Measure on R n Our goal is to construct a notion of the volume, or Lebesgue measure, of rather general subsets of R n that reduces to the usual volume of elementary geometrical sets
More informationA Crash Course in Topological Groups
A Crash Course in Topological Groups Iian B. Smythe Department of Mathematics Cornell University Olivetti Club November 8, 2011 Iian B. Smythe (Cornell) Topological Groups Nov. 8, 2011 1 / 28 Outline 1
More informationKolmogorov complexity and its applications
CS860, Winter, 2010 Kolmogorov complexity and its applications Ming Li School of Computer Science University of Waterloo http://www.cs.uwaterloo.ca/~mli/cs860.html We live in an information society. Information
More informationKolmogorov-Loveland Randomness and Stochasticity
Kolmogorov-Loveland Randomness and Stochasticity Wolfgang Merkle Institut für Informatik, Ruprecht-Karls-Universität Heidelberg Joseph S. Miller Department of Mathematics, University of Connecticut, Storrs
More informationReminder Notes for the Course on Measures on Topological Spaces
Reminder Notes for the Course on Measures on Topological Spaces T. C. Dorlas Dublin Institute for Advanced Studies School of Theoretical Physics 10 Burlington Road, Dublin 4, Ireland. Email: dorlas@stp.dias.ie
More informationTopic 5 Basics of Probability
Topic 5 Basics of Probability Equally Likely Outcomes and the Axioms of Probability 1 / 13 Outline Equally Likely Outcomes Axioms of Probability Consequences of the Axioms 2 / 13 Introduction A probability
More informationA Note On Comparative Probability
A Note On Comparative Probability Nick Haverkamp and Moritz Schulz Penultimate draft. Please quote from the published version (Erkenntnis 2012). Abstract A possible event always seems to be more probable
More informationCourse 212: Academic Year Section 1: Metric Spaces
Course 212: Academic Year 1991-2 Section 1: Metric Spaces D. R. Wilkins Contents 1 Metric Spaces 3 1.1 Distance Functions and Metric Spaces............. 3 1.2 Convergence and Continuity in Metric Spaces.........
More informationA VERY BRIEF REVIEW OF MEASURE THEORY
A VERY BRIEF REVIEW OF MEASURE THEORY A brief philosophical discussion. Measure theory, as much as any branch of mathematics, is an area where it is important to be acquainted with the basic notions and
More informationThe space of located subsets
The space of located subsets Tatsuji Kawai Universtà di Padova Second CORE meeting, 27 January 2017, LMU 1 / 26 The space of located subsets We are interested in a point-free topology on the located subsets
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 informationV (v i + W i ) (v i + W i ) is path-connected and hence is connected.
Math 396. Connectedness of hyperplane complements Note that the complement of a point in R is disconnected and the complement of a (translated) line in R 2 is disconnected. Quite generally, we claim that
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 informationWe are going to discuss what it means for a sequence to converge in three stages: First, we define what it means for a sequence to converge to zero
Chapter Limits of Sequences Calculus Student: lim s n = 0 means the s n are getting closer and closer to zero but never gets there. Instructor: ARGHHHHH! Exercise. Think of a better response for the instructor.
More informationKolmogorov-Loveland Randomness and Stochasticity
Kolmogorov-Loveland Randomness and Stochasticity Wolfgang Merkle 1 Joseph Miller 2 André Nies 3 Jan Reimann 1 Frank Stephan 4 1 Institut für Informatik, Universität Heidelberg 2 Department of Mathematics,
More informationFoundations of non-commutative probability theory
Foundations of non-commutative probability theory Daniel Lehmann School of Engineering and Center for the Study of Rationality Hebrew University, Jerusalem 91904, Israel June 2009 Abstract Kolmogorov s
More informationLECTURE 1. 1 Introduction. 1.1 Sample spaces and events
LECTURE 1 1 Introduction The first part of our adventure is a highly selective review of probability theory, focusing especially on things that are most useful in statistics. 1.1 Sample spaces and events
More informationCW complexes. Soren Hansen. This note is meant to give a short introduction to CW complexes.
CW complexes Soren Hansen This note is meant to give a short introduction to CW complexes. 1. Notation and conventions In the following a space is a topological space and a map f : X Y between topological
More informationSet, functions and Euclidean space. Seungjin Han
Set, functions and Euclidean space Seungjin Han September, 2018 1 Some Basics LOGIC A is necessary for B : If B holds, then A holds. B A A B is the contraposition of B A. A is sufficient for B: If A holds,
More informationWHY WORD PROBLEMS ARE HARD
WHY WORD PROBLEMS ARE HARD KEITH CONRAD 1. Introduction The title above is a joke. Many students in school hate word problems. We will discuss here a specific math question that happens to be named the
More informationPETER A. CHOLAK, PETER GERDES, AND KAREN LANGE
D-MAXIMAL SETS PETER A. CHOLAK, PETER GERDES, AND KAREN LANGE Abstract. Soare [23] proved that the maximal sets form an orbit in E. We consider here D-maximal sets, generalizations of maximal sets introduced
More information