# Extension of measure

Save this PDF as:

Size: px
Start display at page:

## 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.

### Set theory, and set operations

1 Set theory, and set operations Sayan Mukherjee Motivation It goes without saying that a Bayesian statistician should know probability theory in depth to model. Measure theory is not needed unless we

### 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

### Chapter 1. Sigma-Algebras. 1.1 Definition

Chapter 1 Sigma-Algebras 1.1 Definition Consider a set X. A σ algebra F of subsets of X is a collection F of subsets of X satisfying the following conditions: (a) F (b) if B F then its complement B c is

### Notes 2 for Honors Probability and Statistics

Notes 2 for Honors Probability and Statistics Ernie Croot August 24, 2010 1 Examples of σ-algebras and Probability Measures So far, the only examples of σ-algebras we have seen are ones where the sample

### NOTES ON MEASURE THEORY. M. Papadimitrakis Department of Mathematics University of Crete. Autumn of 2004

NOTES ON MEASURE THEORY M. Papadimitrakis Department of Mathematics University of Crete Autumn of 2004 2 Contents 1 σ-algebras 7 1.1 σ-algebras............................... 7 1.2 Generated σ-algebras.........................

### 3 σ-algebras. Solutions to Problems

3 σ-algebras. Solutions to Problems 3.1 3.12 Problem 3.1 (i It is clearly enough to show that A, B A = A B A, because the case of N sets follows from this by induction, the induction step being A 1...

### Chapter 1. Probability Theory: Introduction. Definitions Algebra. RS Chapter 1 Random Variables

Chapter 1 Probability Theory: Introduction Definitions Algebra Definitions: Semiring A collection of sets F is called a semiring if it satisfies: Φ F. If A, B F, then A B F. If A, B F, then there exists

### Solutions to Tutorial 2 (Week 3)

THE UIVERSITY OF SYDEY SCHOOL OF MATHEMATICS AD STATISTICS Solutions to Tutorial 2 (Week 3) MATH3969: Measure Theory and Fourier Analysis (Advanced) Semester 2, 2016 Web Page: http://sydney.edu.au/science/maths/u/ug/sm/math3969/

### Chapter 1. Open Sets, Closed Sets, and Borel Sets

1.4. Borel Sets 1 Chapter 1. Open Sets, Closed Sets, and Borel Sets Section 1.4. Borel Sets Note. Recall that a set of real numbers is open if and only if it is a countable disjoint union of open intervals.

### Structure of Measurable Sets

Structure of Measurable Sets In these notes we discuss the structure of Lebesgue measurable subsets of R from several different points of view. Along the way, we will see several alternative characterizations

### If the sets {A i } are pairwise disjoint, then µ( i=1 A i) =

Note: A standing homework assignment for students in MAT1501 is: let me know about any mistakes, misprints, ambiguities etc that you find in these notes. 1. measures and measurable sets If X is a set,

### Constructions and extensions of measures

CHAPTER 1 Constructions and extensions of measures I compiled these lectures not assuming from the reader any knowledge other than is found in the under-graduate programme of all departments; I can even

### 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

### General theory of stochastic processes

CHAPTER 1 General theory of stochastic processes 1.1. Definition of stochastic process First let us recall the definition of a random variable. A random variable is a random number appearing as a result

### 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.

### Lecture 4: Probability Spaces

EE5110: Probability Foundations for Electrical Engineers July-November 2015 Lecture 4: Probability Spaces Lecturer: Dr. Krishna Jagannathan Scribe: Jainam Doshi, Arjun Nadh and Ajay M 4.1 Introduction

### 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

### Probability Foundations for Electrical Engineers Prof. Krishna Jagannathan Department of Electrical Engineering Indian Institute of Technology, Madras

Probability Foundations for Electrical Engineers Prof. Krishna Jagannathan Department of Electrical Engineering Indian Institute of Technology, Madras Lecture - 9 Borel Sets and Lebesgue Measure -1 (Refer

### Class Notes for MATH 355.

Class Notes for MATH 355. by S. W. Drury Copyright c 2002 2007, by S. W. Drury. Contents Starred chapters and sections should be omitted on a first reading. Double starred sections should be omitted on

### Infinite product spaces

Chapter 7 Infinite product spaces So far we have talked a lot about infinite sequences of independent random variables but we have never shown how to construct such sequences within the framework of probability/measure

### Exercise 1. Let E be a given set. Prove that the following statements are equivalent.

Real Variables, Fall 2014 Problem set 1 Solution suggestions Exercise 1. Let E be a given set. Prove that the following statements are equivalent. (i) E is measurable. (ii) Given ε > 0, there exists an

### Chap2: The Real Number System (See Royden pp40)

Chap2: The Real Number System (See Royden pp40) 1 Open and Closed Sets of Real Numbers The simplest sets of real numbers are the intervals. We define the open interval (a, b) to be the set (a, b) = {x

### Math 563 Measure Theory Project 1 (Funky Functions Group) Luis Zerón, Sergey Dyachenko

Math 563 Measure Theory Project (Funky Functions Group) Luis Zerón, Sergey Dyachenko 34 Let C and C be any two Cantor sets (constructed in Exercise 3) Show that there exists a function F: [,] [,] with

### BOREL SETS, WELL-ORDERINGS OF R AND THE CONTINUUM HYPOTHESIS. Proof. We shall define inductively a decreasing sequence of infinite subsets of N

BOREL SETS, WELL-ORDERINGS OF R AND THE CONTINUUM HYPOTHESIS SIMON THOMAS 1. The Finite Basis Problem Definition 1.1. Let C be a class of structures. Then a basis for C is a collection B C such that for

### Problem Set. Problem Set #2. Math 5322, Fall December 3, 2001 ANSWERS

Problem Set Problem Set #2 Math 5322, Fall 2001 December 3, 2001 ANSWERS i Problem 1. [Problem 18, page 32] Let A P(X) be an algebra, A σ the collection of countable unions of sets in A, and A σδ the collection

### STAT 571 Assignment 1 solutions

STAT 571 Assignment 1 solutions 1. If Ω is a set and C a collection of subsets of Ω, let A be the intersection of all σ-algebras that contain C. rove that A is the σ-algebra generated by C. Solution: Let

### Measuring sets with translation invariant Borel measures

Measuring sets with translation invariant Borel measures University of Warwick Fractals and Related Fields III, 21 September 2015 Measured sets Definition A Borel set A R is called measured if there is

### Descriptive Set Theory

Martin Goldstern Institute of Discrete Mathematics and Geometry Vienna University of Technology Ljubljana, August 2011 Polish spaces Polish space = complete metric separable. Examples N = ω = {0, 1, 2,...}.

### Math 507/420 Homework Assignment #1: Due in class on Friday, September 20. SOLUTIONS

Math 507/420 Homework Assignment #1: Due in class on Friday, September 20. SOLUTIONS 1. Show that a nonempty collection A of subsets is an algebra iff 1) for all A, B A, A B A and 2) for all A A, A c A.

### x = (x 1,..., x d ), y = (y 1,..., y d ) x + y = (x 1 + y 1,..., x d + y d ), x = (x 1,..., x d ), t R tx = (tx 1,..., tx d ).

1. Lebesgue Measure 2. Outer Measure 3. Radon Measures on R 4. Measurable Functions 5. Integration 6. Coin-Tossing Measure 7. Product Measures 8. Change of Coordinates 1. Lebesgue Measure Throughout R

### Measure Theory and Probability

Chapter 2 Measure Theory and Probability 2.1 Introduction In advanced probability courses, a random variable (r.v.) is defined rigorously through measure theory, which is an in-depth mathematics discipline

### 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

### MAA6616 COURSE NOTES FALL 2012

MAA6616 COURSE NOTES FALL 2012 1. σ-algebras Let X be a set, and let 2 X denote the set of all subsets of X. We write E c for the complement of E in X, and for E, F X, write E \ F = E F c. Let E F denote

### 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

### Basic Measure Theory

Basic Measure Theory Heinrich v. Weizsäcker Revised Translation 2004/2008 Fachbereich Mathematik Technische Universität Kaiserslautern 2 Measure Theory (October 20, 2008) Preface The aim of this text is

### 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

### CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE

CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify

### Polish spaces and standard Borel spaces

APPENDIX A Polish spaces and standard Borel spaces We present here the basic theory of Polish spaces and standard Borel spaces. Standard references for this material are the books [143, 231]. A.1. Polish

### Math 317 HW #7 Solutions

Math 17 HW #7 Solutions 1. Exercise..5. Decide which of the following sets are compact. For those that are not compact, show how Definition..1 breaks down. In other words, give an example of a sequence

### Real Analysis. Armin Rainer

Real Analysis Armin Rainer Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern- Platz 1, A-19 Wien, Austria E-mail address: armin.rainer@univie.ac.at Preface These are lecture notes for the course

### Real analysis for graduate students

Real analysis for graduate students Second edition Richard F. Bass ii c Copyright 2013 Richard F. Bass All rights reserved. ISBN-13: 978-1481869140 ISBN-10: 1481869140 First edition published 2011. iii

### 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

### 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

### REAL ANALYSIS I HOMEWORK 2

REAL ANALYSIS I HOMEWORK 2 CİHAN BAHRAN The questions are from Stein and Shakarchi s text, Chapter 1. 1. Prove that the Cantor set C constructed in the text is totally disconnected and perfect. In other

### Sigma Field Notes for Economics 770 : Econometric Theory

1 Sigma Field Notes for Economics 770 : Econometric Theory Jonathan B. Hill Dept. of Economics, University of North Carolina 0. DEFINITIONS AND CONVENTIONS The following is a small list of repeatedly used

### Lecture 2 : Basics of Probability Theory

Lecture 2 : Basics of Probability Theory When an experiment is performed, the realization of the experiment is an outcome in the sample space. If the experiment is performed a number of times, different

### This chapter describes set theory, a mathematical theory that underlies all of modern mathematics.

Appendix A Set Theory This chapter describes set theory, a mathematical theory that underlies all of modern mathematics. A.1 Basic Definitions Definition A.1.1. A set is an unordered collection of elements.

### Analysis I. 1 Measure theory. Piotr Haj lasz. 1.1 σ-algebra.

Analysis I Piotr Haj lasz 1 Measure theory 1.1 σ-algebra. Definition. Let be a set. A collection M of subsets of is σ-algebra if M has the following properties 1. M; 2. A M = \ A M; 3. A 1, A 2, A 3,...

### Math 421, Homework #5 Solutions

Math 421, Homework #5 Solutions (1) (8.3.6) Suppose that E R n and C is a subset of E. (a) Prove that if E is closed, then C is relatively closed in E if and only if C is a closed set (as defined in Definition

### MATH41112/61112 Ergodic Theory Lecture Measure spaces

8. Measure spaces 8.1 Background In Lecture 1 we remarked that ergodic theory is the study of the qualitative distributional properties of typical orbits of a dynamical system and that these properties

### Sets and Cardinality Notes for C. F. Miller

Sets and Cardinality Notes for 620-111 C. F. Miller Semester 1, 2000 Abstract These lecture notes were compiled in the Department of Mathematics and Statistics in the University of Melbourne for the use

### Finite Sets. Theorem 5.1. Two non-empty finite sets have the same cardinality if and only if they are equivalent.

MATH 337 Cardinality Dr. Neal, WKU We now shall prove that the rational numbers are a countable set while R is uncountable. This result shows that there are two different magnitudes of infinity. But we

### 1. Sigma algebras. Let A be a collection of subsets of some fixed set Ω. It is called a σ -algebra with the unit element Ω if. lim sup. j E j = E.

Real Analysis, course outline Denis Labutin 1 Measure theory I 1. Sigma algebras. Let A be a collection of subsets of some fixed set Ω. It is called a σ -algebra with the unit element Ω if (a), Ω A ; (b)

### 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).

### 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

### 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

### A Measure Theory Tutorial (Measure Theory for Dummies)

A Measure Theory Tutorial (Measure Theory for Dummies) Maya R. Gupta {gupta}@ee.washington.edu Dept of EE, University of Washington Seattle WA, 98195-2500 UWEE Technical Report Number UWEETR-2006-0008

### 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

### 2 n = n=1 a n is convergent and we let. i=1

Lecture 4 : Series So far our definition of a sum of numbers applies only to adding a finite set of numbers. We can extend this to a definition of a sum of an infinite set of numbers in much the same way

### Random Variables and Measurable Functions.

Chapter 3 Random Variables and Measurable Functions. 3.1 Measurability Definition 42 (Measurable function) Let f be a function from a measurable space (Ω, F) into the real numbers. We say that the function

### MATH INTRODUCTION TO PROBABILITY FALL 2011

MATH 7550-01 INTRODUCTION TO PROBABILITY FALL 2011 Lecture 4. Measurable functions. Random variables. I have told you that we can work with σ-algebras generated by classes of sets in some indirect ways.

### PART I. THE REAL NUMBERS

PART I. THE REAL NUMBERS This material assumes that you are already familiar with the real number system and the representation of the real numbers as points on the real line. I.1. THE NATURAL NUMBERS

### Probability Theory: STAT310/MATH230; June 7, Amir Dembo

Probability Theory: STAT310/MATH230; June 7, 2012 Amir Dembo E-mail address: amir@math.stanford.edu Department of Mathematics, Stanford University, Stanford, CA 94305. Contents Preface 5 Chapter 1. Probability,

### ) + ˆf (n) sin( 2πnt. = 2 u x 2, t > 0, 0 < x < 1. u(0, t) = u(1, t) = 0, t 0. (x, 0) = 0 0 < x < 1.

Introduction to Fourier analysis This semester, we re going to study various aspects of Fourier analysis. In particular, we ll spend some time reviewing and strengthening the results from Math 425 on Fourier

### 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

### 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

### 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

### Definition of Random Variable A random variable is a function from a sample space S into the real numbers.

.4 Random Variable Motivation example In an opinion poll, we might decide to ask 50 people whether they agree or disagree with a certain issue. If we record a for agree and 0 for disagree, the sample space

### 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

### Assignment 7; Due Friday, November 11

Assignment 7; Due Friday, November 9.8 a The set Q is not connected because we can write it as a union of two nonempty disjoint open sets, for instance U = (, 2) and V = ( 2, ). The connected subsets are

### Sets and Subsets. Countable and Uncountable

Sets and Subsets Countable and Uncountable Reading Appendix A Section A.6.8 Pages 788-792 BIG IDEAS Themes 1. There exist functions that cannot be computed in Java or any other computer language. 2. There

### Lecture 4: Random Variables

Lecture 4: Random Variables 1. Definition of Random variables 1.1 Measurable functions and random variables 1.2 Reduction of the measurability condition 1.3 Transformation of random variables 1.4 σ-algebra

### Analytic Sets Vadim Kulikov Presentation 11 March 2008

Analytic Sets Vadim Kulikov Presentation 11 March 2008 Abstract. The continuous images of Borel sets are not usually Borel sets. However they are measurable and that is the main purpose of this lecture.

### Math 6710 lecture notes

Math 6710 lecture notes Nate Eldredge November 29, 2012 Caution! These lecture notes are very rough. They are mainly intended for my own use during lecture. They almost surely contain errors and typos,

### NONMEASURABLE ALGEBRAIC SUMS OF SETS OF REALS

NONMEASURABLE ALGEBRAIC SUMS OF SETS OF REALS MARCIN KYSIAK Abstract. We present a theorem which generalizes some known theorems on the existence of nonmeasurable (in various senses) sets of the form X+Y.

### 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

### Weak topologies. David Lecomte. May 23, 2006

Weak topologies David Lecomte May 3, 006 1 Preliminaries from general topology In this section, we are given a set X, a collection of topological spaces (Y i ) i I and a collection of maps (f i ) i I such

### 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

### Chapter 6 Finite sets and infinite sets. Copyright 2013, 2005, 2001 Pearson Education, Inc. Section 3.1, Slide 1

Chapter 6 Finite sets and infinite sets Copyright 013, 005, 001 Pearson Education, Inc. Section 3.1, Slide 1 Section 6. PROPERTIES OF THE NATURE NUMBERS 013 Pearson Education, Inc.1 Slide Recall that denotes

### 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

### Basics of Probability

Basics of Probability August 27 and September 1, 2009 1 Introduction A phenomena is called random if the exact outcome is uncertain. The mathematical study of randomness is called the theory of probability.

### 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

### 1 Limiting distribution for a Markov chain

Copyright c 2009 by Karl Sigman Limiting distribution for a Markov chain In these Lecture Notes, we shall study the limiting behavior of Markov chains as time n In particular, under suitable easy-to-check

### Compactness in metric spaces

MATHEMATICS 3103 (Functional Analysis) YEAR 2012 2013, TERM 2 HANDOUT #2: COMPACTNESS OF METRIC SPACES Compactness in metric spaces The closed intervals [a, b] of the real line, and more generally the

### 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.

### An example of a capacity for which all positive Borel sets are thick

An example of a capacity for which all positive Borel sets are thick Micha l Morayne, Szymon Żeberski Wroc law University of Technology Winter School in Abstract Analysis 2016 section Set Theory & Topology

### 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

### 14 Abian and Kemp Theorem 1. For every cardinal k if the union of k many sets of measure zero is of measure zero, then the union of k many measurable

PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE Nouvelle série tome 52 (66), 1992, 13 17 ON MEASURABILITY OF UNCOUNTABLE UNIONS OF MEASURABLE SETS Alexander Abian and Paula Kemp Abstract. This paper deals exclusively

### Limits and convergence.

Chapter 2 Limits and convergence. 2.1 Limit points of a set of real numbers 2.1.1 Limit points of a set. DEFINITION: A point x R is a limit point of a set E R if for all ε > 0 the set (x ε,x + ε) E is

### 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

### Probability - Part I. Definition : A random experiment is an experiment or a process for which the outcome cannot be predicted with certainty.

Probability - Part I Definition : A random experiment is an experiment or a process for which the outcome cannot be predicted with certainty. Definition : The sample space (denoted S) of a random experiment

### 6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, )

6. Metric spaces In this section we review the basic facts about metric spaces. Definitions. A metric on a non-empty set X is a map with the following properties: d : X X [0, ) (i) If x, y X are points

### MAT 235A / 235B: Probability

MAT 235A / 235B: Probability Instructor: Prof. Roman Vershynin 2007-2008 Contents 0.1 Generating σ-algebras........................ 8 1 Probability Measures 9 1.1 Continuity of Probability Measures.................

### 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

### Mathematical foundations of Econometrics

Mathematical foundations of Econometrics G.Gioldasis, UniFe & prof. A.Musolesi, UniFe April 7, 2016.Gioldasis, UniFe & prof. A.Musolesi, UniFe Mathematical foundations of Econometrics April 7, 2016 1 /

### MAT2400 Analysis I. A brief introduction to proofs, sets, and functions

MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take

### Convergent Sequence. Definition A sequence {p n } in a metric space (X, d) is said to converge if there is a point p X with the following property:

Convergent Sequence Definition A sequence {p n } in a metric space (X, d) is said to converge if there is a point p X with the following property: ( ɛ > 0)( N)( n > N)d(p n, p) < ɛ In this case we also

### Random Union, Intersection and Membership

Random Union, Intersection and Membership Douglas Cenzer Department of Mathematics University of Florida June 2009 Logic, Computability and Randomness Luminy, Marseille, France Random Reals Introduction