Extension of measure

Size: px
Start display at page:

Download "Extension of measure"

Transcription

1 1 Extension of measure Sayan Mukherjee Dynkin s π λ theorem We will soon need to define probability measures on infinite and possible uncountable sets, like the power set of the naturals. This is hard. It is easier to define the measure on a much smaller collection of the power set and state consistency conditions that allow us to extend the measure to the the power set. This is what Dynkin s π λ theorem was designed to do. Definition π-system) Given a set Ω a π system is a collection of subsets P that are closed under finite intersections. 1) P is non-empty; 2) A B P whenever A, B P. Definition λ-system) Given a set Ω a λ system is a collection of subsets L that contains Ω and is closed under complementation and disjoint countable unions. 1) Ω L ; 2) A L A c L ; 3) A n L, n 1 with A i A j = i j n=1 A n L. Definition σ-algebra) Let F be a collection of subsets of Ω. F is called a field algebra) if Ω F and F is closed under complementation and countable unions, 1) Ω F; 2) A F A c F; 3) A 1,..., A n F n j=1 A j F; 4) A 1,..., A n,... F j=1 A j F. A σ-algebra is a π system. A σ-algebra is a λ system but a λ system need not be a σ algebra, a λ system is a weaker system. The difference is between unions and disjoint unions. Example Ω = {a, b, c, d} and L = {Ω,, {a, b}, {b, c}, {c, d}, {a, d}, {b, d}, {a, c}} L is closed under disjoint unions but not unions. However: Lemma A class that is both a π system and a λ system is a σ-algebra. Proof. We need to show that the class is closed under countable union. Consider A 1, A 2,..., A n L note that these A i are not disjoint. We disjointify them, B 1 = A 1 and B n = A n \ n 1 A n 1 i = A n Ac i. B n L by π system and complement property of λ system. A i = B i but the {B i } are disjoint so A i = B i L. We will start constructing probability measures on infinite sets and we will need to push the idea of extension of measure. In this the following theorem is central. Theorem Dynkin) 1) If P is a π system and L is a λ system, then P L implies σp) L. b) If P is a π system σp) = L P). Proof of a). Let L 0 be the λ system generated by P, the intersection of all λ systems containing P. This is a λ system and contains P and is contained in every λ system that contains P. So P L 0 L. If L 0 is also a π system, it is a λ system and so is a σ algebra. By minimality of σp) it follows σp) L 0 so P σp) L 0 L. So we need to show that L 0 is a π system. Set A σp) and define G A = {B σp) : A B L P)}.

2 2 1) We show if A L P) then G A is a λ system. i) Ω G A since A Ω = A L P). ii) Suppose B G A. B c A = A\A B) but B G A means A B L P) and since A L P) it holds A \ A B) = B c A L P) by complementarity. Since B c A L P) it holds that B c G A. iii) {B j } are mutually disjoint and B j G A then 2) We show if A P then L P) G A. A B j ) = A B j ) L P). j=1 Since A P L P) from 1) G A is a λ system, j=1 For B P we have A B P since A P and P is a π system. So if B P then A B P L P) implies B G A so P G A and since G A is a λ system L P) G A. 3) We rephrase 2) to state if A P and B L P) then A B L P) if we can extend A to L P) we are done). 4) We show that if A L P) then L P) G A. If B P and A L P) then A B L P) then from 3) it holds A B L P). So when A L P) it holds P G A so from 1) G A is a π system and L P) G A. 5) If A L P) then for any B L P), B G A then A B L P). The following lemma is a result of Dynkin s theorem and highlights uniqueness. Lemma If P i and P 2 are two probability measures on σp) where P is a π system. If P 1 and P 2 agree on P, then they agree on σp). Proof is a λ system and P L so σp) L. L = {A P : P 1 A) = P 2 A)}, We will need to define a probability measure, µ, for all sets in a σ algebra. It will be convenient and almost necessary to define a pre)-measure, µ 0, on a smaller collection F 0 that extends to µ on σf 0 ) and µ is unique, µ 1 F ) = µ 2 F ), F F 0 implies µ 1 F ) = µ 2 F ), F σf 0 ). Restatement of Dynkin s theorem in the above context is that the smallest σ algebra generated from a π system is the λ system generated from that π system. Theorem Let F 0 be a π system then λf 0 ) = σf 0 ). We now define the term extension and develop this idea of extension of measure. The main working example we will use in the development is. Ω = R, S = {a, b] : a b }, Pa, b]) = F b) F a). Remember, that the Borel sets BR) = σs). Goal: we have a probability measure on S we want to extend it to BR). Extension: If G 1 and G 2 are collections of subsets of Ω with G 1 G 2 and P 1 : G 1 [0, 1], P 2 : G 2 [0, 1]. P 2 is an extension of P 1 if P 2 G1 = P 1, P 2 A 1 ) = P 1 A i ) A i G 1.

3 3 Few terms: A measure is P additive if for disjoint sets A 1,..., A n G n A i n n A i G P A i ) = A measure is σ additive if for disjoint sets A 1,..., A n,... G n PA i ). A i G P A i ) = PA i ). Definition Semialgebra) A class S of subsets of Ω is a semialgebra if 1), Ω S 2) S is a π system 3) If A S there exists disjoint sets C 1,..., C n with each C i S such that A c = n C i. Example of Semialgebra: Intervals Ω = R, S = {a, b] : a b }. If I 1, I 2 S then I 1 I 2 S and I c is a union of disjoint intervals. The idea/plan: 1. Start with a semialgebra S 2. Show a unique extension to an algebra AS) generated by S 3. Show a unique extension to σas)) = σs). If S is a semialgebra of Ω then AS) = { S i : I is finite, {S i } is disjoint, S i S}, i I a system of finite collections of mutually disjoint sets. An example of extension of measure for a countable set. Ω = {ω 1,..., ω n,...} i p i p i 0, i = 1 p 1 = 1. F = PΩ) the power set of Ω. We define for all A F the following extension of measure PA) = p i. ω i A For this to be a proper measure the follow need to hold they do) 1. PA) 0 for all A F 2. PΩ) = p i = 1 3. If {A j } with j 1 are disjoint then P Aj ) = j=1 p i = p i = PA j ). w i ja j j ω i A j j Is the power set PΩ) countable? What was the semialgebra above? What would we do if Ω = R? As we consider the following theorems and proofs or proof sketches a good example to keep in mind is the case where Ω = R and S == {a, b] : a < b }.

4 4 Theorem First extension) S is a semialgebra of Ω and P : S [0, 1] is σ-additive on S and PΩ) = 1. There is a unique extension P of P to AS) defined by P S i ) = i PS i ), P Ω) = 1 and P is σ additive on AS). Proof Page 46, Probability Path by Resnick) We need to show that P S i ) = i PS i ), uniquely defines P. Suppose that A AS) has representations A = i I S i = j J S j, we need to verify Since S i A the following hold i I PS i ) = j J PS j). i I PS i ) = i I PS i A) = i I PS i j J = i I P j J S i S j), S j) and S i = j J S i S j S and P is additive on S so and by the same argument PS i ) = PS i S j) = i I i I j J i I PS j) = PS i S j) = j J i I j J i I We now check that P is σ additive on AS). Suppose for i 1, PS i S j), j J PS i S j). j J A i = j J i S ij AS), S ij S, and {A i, i 1} are mutually disjoint and A = A i AS). Since A AS), A can be represented as A = S k, k K S k S, k K, where K is a finite index set. So by the definition of P P A) = k K PS k ). We can write S k = S k A = S k A i = S k S ij. j J i

5 5 Since S k S ij S and j J i S k S ij = S k S and P is σ additive on S PS k ) = PS k S ij ) = PS k S ij ). k K k K j J i k K j J i Note that Sk S ij = A S ij = S ij S, k K and by additivity of P on S we show σ additivity PS k S ij ) = j J i k K = PS ij ) j J i P S ij ) = P A i ). j J i The last step is to check the extension is unique. Assume P 1 and P 2 aretwo additive extensions, then for any A = i I S i AS) we have P 1A) = i I PS i ) = P 2A). We now show how to extend from an algebra to a sigma algebra. Theorem Second extension, Caratheodory) A probability measure P defined on an algebra A has a unique extension to a probability measure P on σa). Proof sketch Page 4, Probability Theory by Varadhan) There are 5 steps to the proof. 1. Define the extension: {A j } are the set of all countable collections of A that are disjoint and we define the outer measure P as P A) = inf A ja j j PA j ). P will be the extension. 2. Show P satisfies properties of a measure i) Countable subattitivity P j A j ) j P A j ). ii) For all A A, P A) = PA). This is done in two steps by showing 1) P A) P A) and 2) P A) PA). 3. Checking measurability of the extension are all of the extra sets in going from the algebra to the σ algebra adding correctly). A set E is measurable if P A) P A E) + P A E c ), for all A A. The class M is the collection of measurable sets. We need to show that M is a σ algebra and P is σ additive on M.

6 6 4. Show A M. This implies that σa) M and P extends from A to σa). 5. Uniqueness. Let P 1 and P 2 be σ additive measures on A then show if for all A A P 1 A) = P 2 A), then P 1 B) = P 2 B), B σa). We now look at two examples. The first is the extension of the Lebesgue measure to Borel sets. The second example illustrates sets that are not measurable. Lebesgue measure: Ω = [0, 1], F = B0, 1]), S = {a, b] : 0 a b 1}. S is a semialgebra and we define the following measure on it λ : S a, b], λ ) = 0, λa, b]) = b a. This measure λ extends to the Borel set B0, 1]). To show that λ extends to B0, 1]) we need to show that it is σ additive on B0, 1]). We first show finite additivity. Set a, b] = K a i, b i ], where a = a 1, b K = b, b i = a i+1. We can see K λa i, b i ] = K b i a i ) = b 1 a 1 + b 2 a 2 + = b K a 1 = b a. We now note a few asymptotic properties of intervals a, b) = [a, b] = {a} = n=1 n=1 n=1 a, b 1 ], a < b n a 1n ], b, a < b a 1n, a ], a < b. The Borel σ algebra contains all singletons a as well as the forms a, b), [a, b], [a, b), a, b]. We now have to address σ additivity. First note that the interval [0, 1] is compact, this means that there exists a finite ε cover of the interval for any ε > 0. Let A n be a sequence of sets from S such that A n. We can define a set B n S such that its closure [B n ] A n and λa n ) λb n ) ε 2 n. Remember A n = and since the sets [B n ] are closed there exists a finite n 0 ε) such that n0 [B n] =. Remember that A n0 A n0 1 A 1 so ) n 0 n0 ) λa n0 ) = λ A n0 \ B k + λ B k n 0 ) = λ A n0 \ B k λ n 0 n0 ) A k \ B k n 0 λa k \ B k ) ε 2 k ε.

7 7 The reason this shows σ additivity is that if for all A n S, A n implies λa n ) 0 then σ additivity holds on σs). Not a measurable set: Let Ω = N be the natural numbers and E be the even numbers and E c the odd numbers and set F = {2 2k + 1,..., 2 2k+1 } = {2,. 5,..., 8, 17,..., 32, 65,..., 128,...}. k=0 1. For n = 2 2k the ratio P n F ) = #[F {1,..., n}]/n = n 1)/3n 1/3 and for n = 2 2k+1, P n F )2n 1)/3n 2/3 so P n cannot converge as n. 2. The even and odd portions of F and F c, A F E and B F c E c, have relative frequencies that do not converge. 3. C = A B does have a relative frequency that converges: P 2n C) = 1/2 + 1/2n so lim n P n = 1/2. 4. Both E and C have well defined asymptotic frequences but A = E C does not and the collection of sets M for which P n converges is not an algebra.

CHAPTER 5. Product Measures

CHAPTER 5. Product Measures 54 CHAPTER 5 Product Measures Given two measure spaces, we may construct a natural measure on their Cartesian product; the prototype is the construction of Lebesgue measure on R 2 as the product of Lebesgue

More information

Our goal first will be to define a product measure on A 1 A 2.

Our goal first will be to define a product measure on A 1 A 2. 1. Tensor product of measures and Fubini theorem. Let (A j, Ω j, µ j ), j = 1, 2, be two measure spaces. Recall that the new σ -algebra A 1 A 2 with the unit element is the σ -algebra generated by the

More information

Basic Measure Theory. 1.1 Classes of Sets

Basic Measure Theory. 1.1 Classes of Sets 1 Basic Measure Theory In this chapter, we introduce the classes of sets that allow for a systematic treatment of events and random observations in the framework of probability theory. Furthermore, we

More information

Math212a1010 Lebesgue measure.

Math212a1010 Lebesgue measure. Math212a1010 Lebesgue measure. October 19, 2010 Today s lecture will be devoted to Lebesgue measure, a creation of Henri Lebesgue, in his thesis, one of the most famous theses in the history of mathematics.

More information

1. Prove that the empty set is a subset of every set.

1. Prove that the empty set is a subset of every set. 1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since

More information

DISINTEGRATION OF MEASURES

DISINTEGRATION OF MEASURES DISINTEGRTION OF MESURES BEN HES Definition 1. Let (, M, λ), (, N, µ) be sigma-finite measure spaces and let T : be a measurable map. (T, µ)-disintegration is a collection {λ y } y of measures on M such

More information

SOLUTIONS TO ASSIGNMENT 1 MATH 576

SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts

More information

Lecture Notes on Measure Theory and Functional Analysis

Lecture Notes on Measure Theory and Functional Analysis Lecture Notes on Measure Theory and Functional Analysis P. Cannarsa & T. D Aprile Dipartimento di Matematica Università di Roma Tor Vergata cannarsa@mat.uniroma2.it daprile@mat.uniroma2.it aa 2006/07 Contents

More information

Ri and. i=1. S i N. and. R R i

Ri and. i=1. S i N. and. R R i The subset R of R n is a closed rectangle if there are n non-empty closed intervals {[a 1, b 1 ], [a 2, b 2 ],..., [a n, b n ]} so that R = [a 1, b 1 ] [a 2, b 2 ] [a n, b n ]. The subset R of R n is an

More information

REAL ANALYSIS LECTURE NOTES: 1.4 OUTER MEASURE

REAL ANALYSIS LECTURE NOTES: 1.4 OUTER MEASURE REAL ANALYSIS LECTURE NOTES: 1.4 OUTER MEASURE CHRISTOPHER HEIL 1.4.1 Introduction We will expand on Section 1.4 of Folland s text, which covers abstract outer measures also called exterior measures).

More information

Elements of probability theory

Elements of probability theory 2 Elements of probability theory Probability theory provides mathematical models for random phenomena, that is, phenomena which under repeated observations yield di erent outcomes that cannot be predicted

More information

The Lebesgue Measure and Integral

The Lebesgue Measure and Integral The Lebesgue Measure and Integral Mike Klaas April 12, 2003 Introduction The Riemann integral, defined as the limit of upper and lower sums, is the first example of an integral, and still holds the honour

More information

Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011

Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely

More information

Basic Probability Concepts

Basic Probability Concepts page 1 Chapter 1 Basic Probability Concepts 1.1 Sample and Event Spaces 1.1.1 Sample Space A probabilistic (or statistical) experiment has the following characteristics: (a) the set of all possible outcomes

More information

Chapter 1 - σ-algebras

Chapter 1 - σ-algebras Page 1 of 17 PROBABILITY 3 - Lecture Notes Chapter 1 - σ-algebras Let Ω be a set of outcomes. We denote by P(Ω) its power set, i.e. the collection of all the subsets of Ω. If the cardinality Ω of Ω is

More information

Probability: Theory and Examples. Rick Durrett. Edition 4.1, April 21, 2013

Probability: Theory and Examples. Rick Durrett. Edition 4.1, April 21, 2013 i Probability: Theory and Examples Rick Durrett Edition 4.1, April 21, 213 Typos corrected, three new sections in Chapter 8. Copyright 213, All rights reserved. 4th edition published by Cambridge University

More information

GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.

GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G. Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete

More information

Mathematics for Econometrics, Fourth Edition

Mathematics for Econometrics, Fourth Edition Mathematics for Econometrics, Fourth Edition Phoebus J. Dhrymes 1 July 2012 1 c Phoebus J. Dhrymes, 2012. Preliminary material; not to be cited or disseminated without the author s permission. 2 Contents

More information

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied

More information

Lecture Note 1 Set and Probability Theory. MIT 14.30 Spring 2006 Herman Bennett

Lecture Note 1 Set and Probability Theory. MIT 14.30 Spring 2006 Herman Bennett Lecture Note 1 Set and Probability Theory MIT 14.30 Spring 2006 Herman Bennett 1 Set Theory 1.1 Definitions and Theorems 1. Experiment: any action or process whose outcome is subject to uncertainty. 2.

More information

An example of a computable

An example of a computable An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept

More information

Lebesgue Measure on R n

Lebesgue 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 information

Math/Stats 425 Introduction to Probability. 1. Uncertainty and the axioms of probability

Math/Stats 425 Introduction to Probability. 1. Uncertainty and the axioms of probability Math/Stats 425 Introduction to Probability 1. Uncertainty and the axioms of probability Processes in the real world are random if outcomes cannot be predicted with certainty. Example: coin tossing, stock

More information

Applications of Methods of Proof

Applications of Methods of Proof CHAPTER 4 Applications of Methods of Proof 1. Set Operations 1.1. Set Operations. The set-theoretic operations, intersection, union, and complementation, defined in Chapter 1.1 Introduction to Sets are

More information

MEASURE AND INTEGRATION. Dietmar A. Salamon ETH Zürich

MEASURE AND INTEGRATION. Dietmar A. Salamon ETH Zürich MEASURE AND INTEGRATION Dietmar A. Salamon ETH Zürich 12 May 2016 ii Preface This book is based on notes for the lecture course Measure and Integration held at ETH Zürich in the spring semester 2014. Prerequisites

More information

PROBABILITY THEORY - PART 1 MEASURE THEORETICAL FRAMEWORK CONTENTS

PROBABILITY THEORY - PART 1 MEASURE THEORETICAL FRAMEWORK CONTENTS PROBABILITY THEORY - PART 1 MEASURE THEORETICAL FRAMEWORK MANJUNATH KRISHNAPUR CONTENTS 1. Discrete probability space 2 2. Uncountable probability spaces? 3 3. Sigma algebras and the axioms of probability

More information

MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

More information

INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS

INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem

More information

University of Miskolc

University of Miskolc University of Miskolc The Faculty of Mechanical Engineering and Information Science The role of the maximum operator in the theory of measurability and some applications PhD Thesis by Nutefe Kwami Agbeko

More information

P (A) = lim P (A) = N(A)/N,

P (A) = lim P (A) = N(A)/N, 1.1 Probability, Relative Frequency and Classical Definition. Probability is the study of random or non-deterministic experiments. Suppose an experiment can be repeated any number of times, so that we

More information

1 if 1 x 0 1 if 0 x 1

1 if 1 x 0 1 if 0 x 1 Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

More information

and s n (x) f(x) for all x and s.t. s n is measurable if f is. REAL ANALYSIS Measures. A (positive) measure on a measurable space

and s n (x) f(x) for all x and s.t. s n is measurable if f is. REAL ANALYSIS Measures. A (positive) measure on a measurable space RAL ANALYSIS A survey of MA 641-643, UAB 1999-2000 M. Griesemer Throughout these notes m denotes Lebesgue measure. 1. Abstract Integration σ-algebras. A σ-algebra in X is a non-empty collection of subsets

More information

E3: PROBABILITY AND STATISTICS lecture notes

E3: PROBABILITY AND STATISTICS lecture notes E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................

More information

Mathematical Methods of Engineering Analysis

Mathematical Methods of Engineering Analysis Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................

More information

CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.

CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,

More information

Point Set Topology. A. Topological Spaces and Continuous Maps

Point Set Topology. A. Topological Spaces and Continuous Maps Point Set Topology A. Topological Spaces and Continuous Maps Definition 1.1 A topology on a set X is a collection T of subsets of X satisfying the following axioms: T 1.,X T. T2. {O α α I} T = α IO α T.

More information

Measure Theory. Jesus Fernandez-Villaverde University of Pennsylvania

Measure Theory. Jesus Fernandez-Villaverde University of Pennsylvania Measure Theory Jesus Fernandez-Villaverde University of Pennsylvania 1 Why Bother with Measure Theory? Kolmogorov (1933). Foundation of modern probability. Deals easily with: 1. Continuous versus discrete

More information

CHAPTER 1 BASIC TOPOLOGY

CHAPTER 1 BASIC TOPOLOGY CHAPTER 1 BASIC TOPOLOGY Topology, sometimes referred to as the mathematics of continuity, or rubber sheet geometry, or the theory of abstract topological spaces, is all of these, but, above all, it is

More information

Probability Theory. Florian Herzog. A random variable is neither random nor variable. Gian-Carlo Rota, M.I.T..

Probability Theory. Florian Herzog. A random variable is neither random nor variable. Gian-Carlo Rota, M.I.T.. Probability Theory A random variable is neither random nor variable. Gian-Carlo Rota, M.I.T.. Florian Herzog 2013 Probability space Probability space A probability space W is a unique triple W = {Ω, F,

More information

GENERIC COMPUTABILITY, TURING DEGREES, AND ASYMPTOTIC DENSITY

GENERIC COMPUTABILITY, TURING DEGREES, AND ASYMPTOTIC DENSITY GENERIC COMPUTABILITY, TURING DEGREES, AND ASYMPTOTIC DENSITY CARL G. JOCKUSCH, JR. AND PAUL E. SCHUPP Abstract. Generic decidability has been extensively studied in group theory, and we now study it in

More information

IAM 530 ELEMENTS OF PROBABILITY AND STATISTICS INTRODUCTION

IAM 530 ELEMENTS OF PROBABILITY AND STATISTICS INTRODUCTION IAM 530 ELEMENTS OF PROBABILITY AND STATISTICS INTRODUCTION 1 WHAT IS STATISTICS? Statistics is a science of collecting data, organizing and describing it and drawing conclusions from it. That is, statistics

More information

Sample Induction Proofs

Sample Induction Proofs Math 3 Worksheet: Induction Proofs III, Sample Proofs A.J. Hildebrand Sample Induction Proofs Below are model solutions to some of the practice problems on the induction worksheets. The solutions given

More information

INTRODUCTORY SET THEORY

INTRODUCTORY SET THEORY M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,

More information

SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties

SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces

More information

The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992), 33 37) Bart de Smit

The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992), 33 37) Bart de Smit The fundamental group of the Hawaiian earring is not free Bart de Smit The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992),

More information

Introduction to Topology

Introduction to Topology Introduction to Topology Tomoo Matsumura November 30, 2010 Contents 1 Topological spaces 3 1.1 Basis of a Topology......................................... 3 1.2 Comparing Topologies.......................................

More information

Chapter 1. Metric Spaces. Metric Spaces. Examples. Normed linear spaces

Chapter 1. Metric Spaces. Metric Spaces. Examples. Normed linear spaces Chapter 1. Metric Spaces Metric Spaces MA222 David Preiss d.preiss@warwick.ac.uk Warwick University, Spring 2008/2009 Definitions. A metric on a set M is a function d : M M R such that for all x, y, z

More information

{f 1 (U), U F} is an open cover of A. Since A is compact there is a finite subcover of A, {f 1 (U 1 ),...,f 1 (U n )}, {U 1,...

{f 1 (U), U F} is an open cover of A. Since A is compact there is a finite subcover of A, {f 1 (U 1 ),...,f 1 (U n )}, {U 1,... 44 CHAPTER 4. CONTINUOUS FUNCTIONS In Calculus we often use arithmetic operations to generate new continuous functions from old ones. In a general metric space we don t have arithmetic, but much of it

More information

Metric Spaces. Chapter 1

Metric Spaces. Chapter 1 Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence

More information

Markov random fields and Gibbs measures

Markov random fields and Gibbs measures Chapter Markov random fields and Gibbs measures 1. Conditional independence Suppose X i is a random element of (X i, B i ), for i = 1, 2, 3, with all X i defined on the same probability space (.F, P).

More information

Cardinality. The set of all finite strings over the alphabet of lowercase letters is countable. The set of real numbers R is an uncountable set.

Cardinality. The set of all finite strings over the alphabet of lowercase letters is countable. The set of real numbers R is an uncountable set. Section 2.5 Cardinality (another) Definition: The cardinality of a set A is equal to the cardinality of a set B, denoted A = B, if and only if there is a bijection from A to B. If there is an injection

More information

Statistics in Geophysics: Introduction and Probability Theory

Statistics in Geophysics: Introduction and Probability Theory Statistics in Geophysics: Introduction and Steffen Unkel Department of Statistics Ludwig-Maximilians-University Munich, Germany Winter Term 2013/14 1/32 What is Statistics? Introduction Statistics is the

More information

Notes: Chapter 2 Section 2.2: Proof by Induction

Notes: Chapter 2 Section 2.2: Proof by Induction Notes: Chapter 2 Section 2.2: Proof by Induction Basic Induction. To prove: n, a W, n a, S n. (1) Prove the base case - S a. (2) Let k a and prove that S k S k+1 Example 1. n N, n i = n(n+1) 2. Example

More information

Automata and Computability. Solutions to Exercises

Automata and Computability. Solutions to Exercises Automata and Computability Solutions to Exercises Fall 25 Alexis Maciel Department of Computer Science Clarkson University Copyright c 25 Alexis Maciel ii Contents Preface vii Introduction 2 Finite Automata

More information

We give a basic overview of the mathematical background required for this course.

We give a basic overview of the mathematical background required for this course. 1 Background We give a basic overview of the mathematical background required for this course. 1.1 Set Theory We introduce some concepts from naive set theory (as opposed to axiomatic set theory). The

More information

God created the integers and the rest is the work of man. (Leopold Kronecker, in an after-dinner speech at a conference, Berlin, 1886)

God created the integers and the rest is the work of man. (Leopold Kronecker, in an after-dinner speech at a conference, Berlin, 1886) Chapter 2 Numbers God created the integers and the rest is the work of man. (Leopold Kronecker, in an after-dinner speech at a conference, Berlin, 1886) God created the integers and the rest is the work

More information

9 More on differentiation

9 More on differentiation Tel Aviv University, 2013 Measure and category 75 9 More on differentiation 9a Finite Taylor expansion............... 75 9b Continuous and nowhere differentiable..... 78 9c Differentiable and nowhere monotone......

More information

Follow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu

Follow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu COPYRIGHT NOTICE: Ariel Rubinstein: Lecture Notes in Microeconomic Theory is published by Princeton University Press and copyrighted, c 2006, by Princeton University Press. All rights reserved. No part

More information

I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I. GROUPS: BASIC DEFINITIONS AND EXAMPLES I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

More information

POWER SETS AND RELATIONS

POWER SETS AND RELATIONS POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty

More information

The sample space for a pair of die rolls is the set. The sample space for a random number between 0 and 1 is the interval [0, 1].

The sample space for a pair of die rolls is the set. The sample space for a random number between 0 and 1 is the interval [0, 1]. Probability Theory Probability Spaces and Events Consider a random experiment with several possible outcomes. For example, we might roll a pair of dice, flip a coin three times, or choose a random real

More information

+ Section 6.2 and 6.3

+ Section 6.2 and 6.3 Section 6.2 and 6.3 Learning Objectives After this section, you should be able to DEFINE and APPLY basic rules of probability CONSTRUCT Venn diagrams and DETERMINE probabilities DETERMINE probabilities

More information

This chapter is all about cardinality of sets. At first this looks like a

This chapter is all about cardinality of sets. At first this looks like a CHAPTER Cardinality of Sets This chapter is all about cardinality of sets At first this looks like a very simple concept To find the cardinality of a set, just count its elements If A = { a, b, c, d },

More information

FIXED POINT SETS OF FIBER-PRESERVING MAPS

FIXED POINT SETS OF FIBER-PRESERVING MAPS FIXED POINT SETS OF FIBER-PRESERVING MAPS Robert F. Brown Department of Mathematics University of California Los Angeles, CA 90095 e-mail: rfb@math.ucla.edu Christina L. Soderlund Department of Mathematics

More information

Automata on Infinite Words and Trees

Automata on Infinite Words and Trees Automata on Infinite Words and Trees Course notes for the course Automata on Infinite Words and Trees given by Dr. Meghyn Bienvenu at Universität Bremen in the 2009-2010 winter semester Last modified:

More information

Notes on metric spaces

Notes on metric spaces Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.

More information

Omega Automata: Minimization and Learning 1

Omega Automata: Minimization and Learning 1 Omega Automata: Minimization and Learning 1 Oded Maler CNRS - VERIMAG Grenoble, France 2007 1 Joint work with A. Pnueli, late 80s Summary Machine learning in general and of formal languages in particular

More information

People have thought about, and defined, probability in different ways. important to note the consequences of the definition:

People have thought about, and defined, probability in different ways. important to note the consequences of the definition: PROBABILITY AND LIKELIHOOD, A BRIEF INTRODUCTION IN SUPPORT OF A COURSE ON MOLECULAR EVOLUTION (BIOL 3046) Probability The subject of PROBABILITY is a branch of mathematics dedicated to building models

More information

The Dirichlet Unit Theorem

The Dirichlet Unit Theorem Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

More information

Dedekind s forgotten axiom and why we should teach it (and why we shouldn t teach mathematical induction in our calculus classes)

Dedekind s forgotten axiom and why we should teach it (and why we shouldn t teach mathematical induction in our calculus classes) Dedekind s forgotten axiom and why we should teach it (and why we shouldn t teach mathematical induction in our calculus classes) by Jim Propp (UMass Lowell) March 14, 2010 1 / 29 Completeness Three common

More information

The Banach-Tarski Paradox

The Banach-Tarski Paradox University of Oslo MAT2 Project The Banach-Tarski Paradox Author: Fredrik Meyer Supervisor: Nadia S. Larsen Abstract In its weak form, the Banach-Tarski paradox states that for any ball in R, it is possible

More information

1 Norms and Vector Spaces

1 Norms and Vector Spaces 008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)

More information

Minimal R 1, minimal regular and minimal presober topologies

Minimal R 1, minimal regular and minimal presober topologies Revista Notas de Matemática Vol.5(1), No. 275, 2009, pp.73-84 http://www.saber.ula.ve/notasdematematica/ Comisión de Publicaciones Departamento de Matemáticas Facultad de Ciencias Universidad de Los Andes

More information

Notes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand

Notes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand Notes V General Equilibrium: Positive Theory In this lecture we go on considering a general equilibrium model of a private ownership economy. In contrast to the Notes IV, we focus on positive issues such

More information

The Henstock-Kurzweil-Stieltjes type integral for real functions on a fractal subset of the real line

The Henstock-Kurzweil-Stieltjes type integral for real functions on a fractal subset of the real line The Henstock-Kurzweil-Stieltjes type integral for real functions on a fractal subset of the real line D. Bongiorno, G. Corrao Dipartimento di Ingegneria lettrica, lettronica e delle Telecomunicazioni,

More information

Metric Spaces Joseph Muscat 2003 (Last revised May 2009)

Metric Spaces Joseph Muscat 2003 (Last revised May 2009) 1 Distance J Muscat 1 Metric Spaces Joseph Muscat 2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) 1 Distance A metric space can be thought of

More information

MATHEMATICAL METHODS OF STATISTICS

MATHEMATICAL METHODS OF STATISTICS MATHEMATICAL METHODS OF STATISTICS By HARALD CRAMER TROFESSOK IN THE UNIVERSITY OF STOCKHOLM Princeton PRINCETON UNIVERSITY PRESS 1946 TABLE OF CONTENTS. First Part. MATHEMATICAL INTRODUCTION. CHAPTERS

More information

LECTURE NOTES IN MEASURE THEORY. Christer Borell Matematik Chalmers och Göteborgs universitet 412 96 Göteborg (Version: January 12)

LECTURE NOTES IN MEASURE THEORY. Christer Borell Matematik Chalmers och Göteborgs universitet 412 96 Göteborg (Version: January 12) 1 LECTURE NOTES IN MEASURE THEORY Christer Borell Matematik Chalmers och Göteborgs universitet 412 96 Göteborg (Version: January 12) 2 PREFACE These are lecture notes on integration theory for a eight-week

More information

Lemma 5.2. Let S be a set. (1) Let f and g be two permutations of S. Then the composition of f and g is a permutation of S.

Lemma 5.2. Let S be a set. (1) Let f and g be two permutations of S. Then the composition of f and g is a permutation of S. Definition 51 Let S be a set bijection f : S S 5 Permutation groups A permutation of S is simply a Lemma 52 Let S be a set (1) Let f and g be two permutations of S Then the composition of f and g is a

More information

Today s Topics. Primes & Greatest Common Divisors

Today s Topics. Primes & Greatest Common Divisors Today s Topics Primes & Greatest Common Divisors Prime representations Important theorems about primality Greatest Common Divisors Least Common Multiples Euclid s algorithm Once and for all, what are prime

More information

INTRODUCTION TO TOPOLOGY

INTRODUCTION TO TOPOLOGY INTRODUCTION TO TOPOLOGY ALEX KÜRONYA In preparation January 24, 2010 Contents 1. Basic concepts 1 2. Constructing topologies 13 2.1. Subspace topology 13 2.2. Local properties 18 2.3. Product topology

More information

Poisson process. Etienne Pardoux. Aix Marseille Université. Etienne Pardoux (AMU) CIMPA, Ziguinchor 1 / 8

Poisson process. Etienne Pardoux. Aix Marseille Université. Etienne Pardoux (AMU) CIMPA, Ziguinchor 1 / 8 Poisson process Etienne Pardoux Aix Marseille Université Etienne Pardoux (AMU) CIMPA, Ziguinchor 1 / 8 The standard Poisson process Let λ > be given. A rate λ Poisson (counting) process is defined as P

More information

Math 104: Introduction to Analysis

Math 104: Introduction to Analysis Math 104: Introduction to Analysis Evan Chen UC Berkeley Notes for the course MATH 104, instructed by Charles Pugh. 1 1 August 29, 2013 Hard: #22 in Chapter 1. Consider a pile of sand principle. You wish

More information

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear

More information

Operator-valued version of conditionally free product

Operator-valued version of conditionally free product STUDIA MATHEMATICA 153 (1) (2002) Operator-valued version of conditionally free product by Wojciech Młotkowski (Wrocław) Abstract. We present an operator-valued version of the conditionally free product

More information

No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics

No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results

More information

Solutions to TOPICS IN ALGEBRA I.N. HERSTEIN. Part II: Group Theory

Solutions to TOPICS IN ALGEBRA I.N. HERSTEIN. Part II: Group Theory Solutions to TOPICS IN ALGEBRA I.N. HERSTEIN Part II: Group Theory No rights reserved. Any part of this work can be reproduced or transmitted in any form or by any means. Version: 1.1 Release: Jan 2013

More information

Example 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum. asin. k, a, and b. We study stability of the origin x

Example 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum. asin. k, a, and b. We study stability of the origin x Lecture 4. LaSalle s Invariance Principle We begin with a motivating eample. Eample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum Dynamics of a pendulum with friction can be written

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

More information

Chapter 3: The basic concepts of probability

Chapter 3: The basic concepts of probability Chapter 3: The basic concepts of probability Experiment: a measurement process that produces quantifiable results (e.g. throwing two dice, dealing cards, at poker, measuring heights of people, recording

More information

x if x 0, x if x < 0.

x if x 0, x if x < 0. Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the

More information

13 Infinite Sets. 13.1 Injections, Surjections, and Bijections. mcs-ftl 2010/9/8 0:40 page 379 #385

13 Infinite Sets. 13.1 Injections, Surjections, and Bijections. mcs-ftl 2010/9/8 0:40 page 379 #385 mcs-ftl 2010/9/8 0:40 page 379 #385 13 Infinite Sets So you might be wondering how much is there to say about an infinite set other than, well, it has an infinite number of elements. Of course, an infinite

More information

n k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +...

n k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +... 6 Series We call a normed space (X, ) a Banach space provided that every Cauchy sequence (x n ) in X converges. For example, R with the norm = is an example of Banach space. Now let (x n ) be a sequence

More information

CODING TRUE ARITHMETIC IN THE MEDVEDEV AND MUCHNIK DEGREES

CODING TRUE ARITHMETIC IN THE MEDVEDEV AND MUCHNIK DEGREES CODING TRUE ARITHMETIC IN THE MEDVEDEV AND MUCHNIK DEGREES PAUL SHAFER Abstract. We prove that the first-order theory of the Medvedev degrees, the first-order theory of the Muchnik degrees, and the third-order

More information

GENERATING SETS KEITH CONRAD

GENERATING SETS KEITH CONRAD GENERATING SETS KEITH CONRAD 1 Introduction In R n, every vector can be written as a unique linear combination of the standard basis e 1,, e n A notion weaker than a basis is a spanning set: a set of vectors

More information

Finite and discrete probability distributions

Finite and discrete probability distributions 8 Finite and discrete probability distributions To understand the algorithmic aspects of number theory and algebra, and applications such as cryptography, a firm grasp of the basics of probability theory

More information

Ultraproducts and Applications I

Ultraproducts and Applications I Ultraproducts and Applications I Brent Cody Virginia Commonwealth University September 2, 2013 Outline Background of the Hyperreals Filters and Ultrafilters Construction of the Hyperreals The Transfer

More information

Probability for Estimation (review)

Probability for Estimation (review) Probability for Estimation (review) In general, we want to develop an estimator for systems of the form: x = f x, u + η(t); y = h x + ω(t); ggggg y, ffff x We will primarily focus on discrete time linear

More information

Reference: Introduction to Partial Differential Equations by G. Folland, 1995, Chap. 3.

Reference: Introduction to Partial Differential Equations by G. Folland, 1995, Chap. 3. 5 Potential Theory Reference: Introduction to Partial Differential Equations by G. Folland, 995, Chap. 3. 5. Problems of Interest. In what follows, we consider Ω an open, bounded subset of R n with C 2

More information