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

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 : a < x < b}. We also consider the infinite interval (a, ), and (a, ) = {x : a < x}, (, b) = {x : x < b}. Sometimes we write (, ) for the set of the all real numbers. We define the closed interval [a, b] to be the set [a, b] = {x : a x b}. For closed intervals we take a and b finite but always assume that a < b. The half-open interval (a, b] is defined to be and (a, b] = {x : a < x b}, [a, b) = {x : a x < b}. A generalization of the notion of an open interval is given by that of an open set. Definition 1 A set O of real numbers is called open if for each x O there is a δ > 0 such that each y with x y < δ belongs to O. Definition 2 A set O of real numbers is called open if for each x O there is an open interval I such that x I O. Example 3 1. The open intervals are examples of open sets. 2. the empty set and the set R of real numbers are open. 1

2 We establish some properties of open sets. Proposition 4 The intersection O 1 O2 of two open sets O 1 and O 2 is open. Proof. Let x O 1 O2. Since x O 1 and O 1 is open, there is a δ 1 > 0 such that all y with x y < δ 1 belongs to O 1. Similarly, there is a δ 2 > 0 such that all y with x y < δ 2 belongs to O 2. Take δ = min {δ 1, δ 2 }. Then δ > 0 and if x y < δ, then y belongs to both O 1 and O 2 which means to O 1 O2. Corollary 5 The intersection of any finite collection of open sets is open. Proposition 6 The union of any collection of open sets is open. Example 7 It is not true that the intersection of any collection of open sets is open. Take for example O n to be the open interval ( 1 n, 1 ). Then n and {0} is not an open set. O n = {0}, n=1 Proposition 8 Every open set of real numbers is the union of a countable collection of disjoint open intervals. Proof. Since O is open, for each x O there is a y > x such that (x, y) O and there is a z < x such that (z, x) O. Let b = sup {y : (x, y) O}, a = inf {z : (z, x) O}. Then a < x < b, and x I x = (a, b) O. Characterization of the b: ɛ > 0, b ɛ < x < b. Characterization of the a: ɛ > 0, a < x < a + ɛ. Which implies that a / O and b / O. Consider the collection of open intervals {I x }, x O. Since each x in O is in I x and each I x is contained in O, we have that O = I x. Let us prove that the open intervals must be disjoint. In fact, let (a, b) and (c, d) be two intervals in this collection with a point in common. Then, we must have that c < b and a < d. Since c / O then c / (a, b) and we have that c a. Since a / O then a / (c, d) and we have that a c. Thus a = c. Similarly, we can prove that b = d. Thus two different intervals in the collection {I x } must be disjoint. It remains only to prove that this collection is countable: Each open interval contains a rational number (Axiom of Archimedes), hence the collection can be put in a one-to-one correspondence with a subset of rational numbers. Thus, this collection is countable. 2

3 Proposition 9 (Lindelöf) Let C a collection of open sets of real numbers. Then there is a countable subcollection {O i } of C such that O C O = O i. Proof. See Royden pp42. We shall also study the notion of a closed set which generalizes the notion of a closed interval. Definition 10 (Point of closure or limit point) A real number x is called a point of closure of a set E if for every δ > 0 there is a y E (x y) such that x y < δ. This equivalent to say that x is a point of closure of E if every open interval containing x also contains a point of E, that is I x (I x E) {x}. We denote the set of points of closure of E by E. Thus E E. Example 11 2 is a point of closure of the interval (2, ) and its closure is equal to [2, ), that is (2, ) = [2, ). i=1 2 is not a point of closure of the set {2} (3, ). Proposition 12 If A B then A B. And A B = A B. Definition 13 (Closed set) A set F is called closed if F = F. (it is sufficient to state that F F ). Example 14 [a, b] and [a, ) are closed sets. Proposition 15 For any set E, the set E is closed, that is E = E. Proposition 16 The complement of an open set is closed and the complement of a closed set is open. Proof. O is open = if x O there exits δ > 0 such that if x y < δ then y O. This means that y / Õ, which means that x cannot be a point of closure of Õ since there no y in Õ with x y < δ. Thus Õ contains all its points of closure and therefore is closed. Proposition 17 The union F 1 F 2 of two closed sets F 1 and F 2 is closed. Proof. F 1 F 2 = F 1 F 2 = F 1 F 2. Proposition 18 The intersection of any collection of closed sets is closed. 3

4 Proof. Using the De Morgan s Law we have that i=1 F i = F i=1 i is open. Hence i=1 F i is closed. We say that a collection C of sets covers a set F if F {O : O C}. In this case the collection C is called a covering of F. If each O is open, we call C an open covering of F. If C contains a finite number of sets, we call C a finite covering. Definition 19 (compact set) A set F is called compact if every open covering of F can be reduced to a finite open covering of F, that is if C is a collection of open sets such that F {O : O C}, then there is a finite collection {O 1,..., O n } of sets in C such that n F O i. i=1 Theorem 20 (Heine-Borel) A subset of R is compact if and only if it is closed and bounded. Example 21 [0, 2] is compact. [2, ) is not compact because it is not bounded. Cantor sets. Cantor sets are fascinating examples of compact sets. Here is how to construct the standard Cantor set: Start with the unit interval [0,1] and remove its open middle third, (1/3,2/3). Then remove the open middle third from the remaining two intervals, and so on. This gives you a nested sequence C 0 C 1 C 2... where C 0 = [0, 1], C 1 is the union of two intervals [0, 1/3] and [2/3, 1], C 2 is the union of four intervals [0, 1/9], [2/9, 1/3], [2/3, 7/9] and [8/9, 1], C 3 is the union of eight intervals and so on. In general C n is the union of 2 n intervals, each of length 1/3 n. The Cantor set denoted by C is given by C := C n. Clearly C contains the endpoints of each of the intervals comprising C n. Actually, it contains uncountably many more points than these endpoints. C is uncountable and compact. 2 Continuous functions Let f be a real-valued function whose domain of definition is a set E of real numbers. Definition 22 We say that f is continuous at the point x E if given ɛ > 0, there is a δ > 0 such that for all y E with x y < δ we have f(x) f(y) < ɛ. 4

5 Definition 23 The function f is said to be continuous on a A E if it is continuous at each point of A. Proposition 24 Let f continuous on a closed and bounded set F E. Then f is bounded on F and assumes its maximum and minimum on F ; that is, there are points x 1, x 2 F such that f(x 1 ) f(x) f(x 2 ), x F. Proof. F is compact this means that from each open covering of F we can extract a finite open sub-covering of F. Let us first prove that f is bounded on F. Since f is continuous on F, this means that for each ɛ > 0, δ > 0, x, y F, x y < δ = f(x) f(y) < ɛ. For ɛ = 1, for each x F, there is an open interval I x x such that f(y) < 1 + f(x) for y I x F. On the other hand, the collection {I x : x F } is an open covering of F which is compact so we can extract a finite subcollection {I x1,..., I xn } such n that F I xi. Let M = 1 + max { f(x 1 ),..., f(x n ) }. i=1 If y F then there exits k in [1, n] such that y in I xk. Hence f(y) < 1 + f(x k ) M. Hence f is bounded on F. Now, we have to prove that f assumes its maximum and minimum on F. Let M := sup {f(x) : x F }. Then because f is bounded M <. We have to prove that there exists x 1 F such that M = f(x 1 ). Suppose not, then f(x) < M x F. By the continuity of f, for all ɛ > 0 there is an open interval 1 I x x such that f(y) < f(x) + ɛ y I x F. Let us take ɛ = (M f(x)) 2 then f(y) < 1 2 (f(x)+m) x I x F. Let a := max {f(x1 ),..., f(x n )} < M. Now using the fact that F is compact, each y F belongs to some I xk f(y) < 1 2 (f(x k) + M) 1 2 (M + a). This means that 1 2 (M + a) is a bound for f. Which contradicts the fact that M is the supremum. Hence there is x 1 F such that f(x 1 = M. Use the similar Proof for the minimum. Proposition 25 Let f be a real-valued function on R. Then f is continuous if an only if for each open set O of real numbers f 1 (O) ia an open set. Proof. (= ): Let ɛ > 0. The interval I := (f(x) ɛ, f(x) + ɛ) is an open interval and so its inverse image must be open. Which means that x f 1 (I), because f 1 (I) := {x R : f(x) I}. Hence there is a δ > 0 such that (x δ, x + δ) f 1 (I), which means that if y (x δ, x + δ) then y f 1 (I). In other words, if x y < δ then and 5

6 f(y) (f(x) ɛ, f(x) + ɛ); that is f(x) f(y) < ɛ. Hence f is continuous, since x is arbitrary. ( =): See Royden. Definition 26 (Uniform continuity) A real-valued function f is defined on a set E is said to be uniformly continuous on E if given ɛ > 0, there is a δ > 0 such that for all x and y in E, if x y < δ then f(x) f(y) < ɛ. Proposition 27 If a real-valued function f is defined and continuous on a compact set F R, then it is uniformly continuous on F. Proof. See Royden pp 48. Definition 28 (Pointwise convergence of sequence of functions) A sequence of functions {f n } defined on a set E is said to converge pointwise on E to a function f if for every x E, lim f n(x) = f(x); that is n x E, ɛ > 0, N N, n N, f(x) f n (x) < ɛ. Here N can depend on ɛ and x. Definition 29 (Uniform convergence of sequence of functions) A sequence of functions {f n } converges uniformly to f on E if ɛ > 0, N(ɛ), n N(ɛ), f(x) f n (x) < ɛ, forall x E. 6

### 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

### {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

### 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

### 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.

### 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

### 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

### 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

### 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

### x a x 2 (1 + x 2 ) n.

Limits and continuity Suppose that we have a function f : R R. Let a R. We say that f(x) tends to the limit l as x tends to a; lim f(x) = l ; x a if, given any real number ɛ > 0, there exists a real number

### 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

### Metric Spaces. Chapter 7. 7.1. Metrics

Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

### 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

### 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

### 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

### 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.

### I. Pointwise convergence

MATH 40 - NOTES Sequences of functions Pointwise and Uniform Convergence Fall 2005 Previously, we have studied sequences of real numbers. Now we discuss the topic of sequences of real valued functions.

### 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).

### 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

### 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

### 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.............................

### Open and Closed Sets

Open and Closed Sets Definition: A subset S of a metric space (X, d) is open if it contains an open ball about each of its points i.e., if x S : ɛ > 0 : B(x, ɛ) S. (1) Theorem: (O1) and X are open sets.

### Practice with Proofs

Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using

### 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

### 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......

### 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

### Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

### 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

### 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.

### 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

### 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

### Finite dimensional topological vector spaces

Chapter 3 Finite dimensional topological vector spaces 3.1 Finite dimensional Hausdorff t.v.s. Let X be a vector space over the field K of real or complex numbers. We know from linear algebra that the

### 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)

### Duality of linear conic problems

Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least

### BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but

### So let us begin our quest to find the holy grail of real analysis.

1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers

### 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

### 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

### 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

### 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

### 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

### 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,

### 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.......................................

### ON A GLOBALIZATION PROPERTY

APPLICATIONES MATHEMATICAE 22,1 (1993), pp. 69 73 S. ROLEWICZ (Warszawa) ON A GLOBALIZATION PROPERTY Abstract. Let (X, τ) be a topological space. Let Φ be a class of realvalued functions defined on X.

### A Problem With The Rational Numbers

Solvability of Equations Solvability of Equations 1. In fields, linear equations ax + b = 0 are solvable. Solvability of Equations 1. In fields, linear equations ax + b = 0 are solvable. 2. Quadratic equations

### 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

### THE BANACH CONTRACTION PRINCIPLE. Contents

THE BANACH CONTRACTION PRINCIPLE ALEX PONIECKI Abstract. This paper will study contractions of metric spaces. To do this, we will mainly use tools from topology. We will give some examples of contractions,

### 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.,

### The Ergodic Theorem and randomness

The Ergodic Theorem and randomness Peter Gács Department of Computer Science Boston University March 19, 2008 Peter Gács (Boston University) Ergodic theorem March 19, 2008 1 / 27 Introduction Introduction

### Extension of measure

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.

### 4.1. Definitions. A set may be viewed as any well defined collection of objects, called elements or members of the set.

Section 4. Set Theory 4.1. Definitions A set may be viewed as any well defined collection of objects, called elements or members of the set. Sets are usually denoted with upper case letters, A, B, X, Y,

### 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

### Recursion Theory in Set Theory

Contemporary Mathematics Recursion Theory in Set Theory Theodore A. Slaman 1. Introduction Our goal is to convince the reader that recursion theoretic knowledge and experience can be successfully applied

### Lecture Notes on Topology for MAT3500/4500 following J. R. Munkres textbook. John Rognes

Lecture Notes on Topology for MAT3500/4500 following J. R. Munkres textbook John Rognes November 29th 2010 Contents Introduction v 1 Set Theory and Logic 1 1.1 ( 1) Fundamental Concepts..............................

### 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

### 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

### 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

### 4. Expanding dynamical systems

4.1. Metric definition. 4. Expanding dynamical systems Definition 4.1. Let X be a compact metric space. A map f : X X is said to be expanding if there exist ɛ > 0 and L > 1 such that d(f(x), f(y)) Ld(x,

### Arbitrage and asset market equilibrium in infinite dimensional economies with risk-averse expected utilities

Arbitrage and asset market equilibrium in infinite dimensional economies with risk-averse expected utilities Thai Ha-Huy, Cuong Le Van and Manh-Hung Nguyen April 22, 2013 Abstract We consider a model with

### 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

### 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),

### 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

### F. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)

Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p -CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p

### Topics in weak convergence of probability measures

This version: February 24, 1999 To be cited as: M. Merkle, Topics in weak convergence of probability measures, Zb. radova Mat. Inst. Beograd, 9(17) (2000), 235-274 Topics in weak convergence of probability

### 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

### 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

### Mikołaj Krupski. Regularity properties of measures on compact spaces. Instytut Matematyczny PAN. Praca semestralna nr 1 (semestr zimowy 2010/11)

Mikołaj Krupski Instytut Matematyczny PAN Regularity properties of measures on compact spaces Praca semestralna nr 1 (semestr zimowy 2010/11) Opiekun pracy: Grzegorz Plebanek Regularity properties of measures

### Convex analysis and profit/cost/support functions

CALIFORNIA INSTITUTE OF TECHNOLOGY Division of the Humanities and Social Sciences Convex analysis and profit/cost/support functions KC Border October 2004 Revised January 2009 Let A be a subset of R m

### (Basic definitions and properties; Separation theorems; Characterizations) 1.1 Definition, examples, inner description, algebraic properties

Lecture 1 Convex Sets (Basic definitions and properties; Separation theorems; Characterizations) 1.1 Definition, examples, inner description, algebraic properties 1.1.1 A convex set In the school geometry

### Discrete Mathematics: Solutions to Homework (12%) For each of the following sets, determine whether {2} is an element of that set.

Discrete Mathematics: Solutions to Homework 2 1. (12%) For each of the following sets, determine whether {2} is an element of that set. (a) {x R x is an integer greater than 1} (b) {x R x is the square

### 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

### 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

### 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

### 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

### Commentationes Mathematicae Universitatis Carolinae

Commentationes Mathematicae Universitatis Carolinae Adam Emeryk; Władysław Kulpa The Sorgenfrey line has no connected compactification Commentationes Mathematicae Universitatis Carolinae, Vol. 18 (1977),

### 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

### Lecture 7: Continuous Random Variables

Lecture 7: Continuous Random Variables 21 September 2005 1 Our First Continuous Random Variable The back of the lecture hall is roughly 10 meters across. Suppose it were exactly 10 meters, and consider

### MA4001 Engineering Mathematics 1 Lecture 10 Limits and Continuity

MA4001 Engineering Mathematics 1 Lecture 10 Limits and Dr. Sarah Mitchell Autumn 2014 Infinite limits If f(x) grows arbitrarily large as x a we say that f(x) has an infinite limit. Example: f(x) = 1 x

### 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

### 1. Let X and Y be normed spaces and let T B(X, Y ).

Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: NVP, Frist. 2005-03-14 Skrivtid: 9 11.30 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok

### HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following

### 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

### INCIDENCE-BETWEENNESS GEOMETRY

INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full

### 6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium

6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium Asu Ozdaglar MIT February 18, 2010 1 Introduction Outline Pricing-Congestion Game Example Existence of a Mixed

### 2 Complex Functions and the Cauchy-Riemann Equations

2 Complex Functions and the Cauchy-Riemann Equations 2.1 Complex functions In one-variable calculus, we study functions f(x) of a real variable x. Likewise, in complex analysis, we study functions f(z)

### Sets and set operations

CS 441 Discrete Mathematics for CS Lecture 7 Sets and set operations Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square asic discrete structures Discrete math = study of the discrete structures used

### Limit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x)

SEC. 4.1 TAYLOR SERIES AND CALCULATION OF FUNCTIONS 187 Taylor Series 4.1 Taylor Series and Calculation of Functions Limit processes are the basis of calculus. For example, the derivative f f (x + h) f

### 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

### 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

ADVANCED CALCULUS Lecture notes for MA 440/540 & 441/541 2015/16 Rudi Weikard log x 1 1 2 3 4 5 x 1 2 Based on lecture notes by G. Stolz and G. Weinstein Version of September 3, 2016 1 Contents First things

### 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

### 1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form

1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and

### Metric Spaces. Lecture Notes and Exercises, Fall 2015. M.van den Berg

Metric Spaces Lecture Notes and Exercises, Fall 2015 M.van den Berg School of Mathematics University of Bristol BS8 1TW Bristol, UK mamvdb@bristol.ac.uk 1 Definition of a metric space. Let X be a set,

### An Introduction to Real Analysis. John K. Hunter. Department of Mathematics, University of California at Davis

An Introduction to Real Analysis John K. Hunter Department of Mathematics, University of California at Davis Abstract. These are some notes on introductory real analysis. They cover the properties of the

### Clicker Question. Theorems/Proofs and Computational Problems/Algorithms MC215: MATHEMATICAL REASONING AND DISCRETE STRUCTURES

MC215: MATHEMATICAL REASONING AND DISCRETE STRUCTURES Tuesday, 1/21/14 General course Information Sets Reading: [J] 1.1 Optional: [H] 1.1-1.7 Exercises: Do before next class; not to hand in [J] pp. 12-14:

### A domain of spacetime intervals in general relativity

A domain of spacetime intervals in general relativity Keye Martin Department of Mathematics Tulane University New Orleans, LA 70118 United States of America martin@math.tulane.edu Prakash Panangaden School

### ALMOST COMMON PRIORS 1. INTRODUCTION

ALMOST COMMON PRIORS ZIV HELLMAN ABSTRACT. What happens when priors are not common? We introduce a measure for how far a type space is from having a common prior, which we term prior distance. If a type

### 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

### a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given