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


 Rudolph Mills
 2 years ago
 Views:
Transcription
1 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 the following properties: i) F is continuous and bijective ii) F is monotically increasing iii) F maps surjectively onto C C [ Hint: Copy the construction of the standard CantorLebesgue function] Proof Let I, I, be the intervals removed from [,] in the construction of and let J, J, be the intervals deleted from [,] in the construction of C, arranged in the same order That is, I and J are the middle intervals removed in the first step; I and J are the left middles and I 3, J 3 are the right middles removed in the second step; and so on Then map the interval I n onto the interval J n linearly and increasingly, for n=,, (see graph below) C J 3 J J I I I 3 Note that [,] \ C is dense in [,] Indeed, take x [,] ; if x [,] \ C there is nothing to prove If x C then by Exercise 4 there is a sequence { x n } n such that x n [,] \ C and x n x Thus any neighborhood of x contains one element of [,] \ C (in fact infinitely many) different from x Hence x is a limit point of [,] \ C This proves that [,] \ C is dense in [,] A similar argument shows that [,] \ C is also dense in [,]
2 Hence F: [,] \ C [,] \ C is defined and strictly increasing on a dense subset of [,] Since the range of F is also dense on [,], the domain of F can be extended to [,] so that F is increasing and continuous on [,] with range [,] We extend the domain of definition of F to all of [,] by putting and F() =, F() =, F(x ) = Sup{ F(: x [,] \ C, x < x} We see that F is now defined on all of [,] since F is bounded on [,] \ C Property (ii) F is monotonically increasing Let x [,] such that x < We must consider several cases: x, x a) Both and are in [,] \ C Then, by construction, F x ) < F( ) x x x C b) is in [,] \ and x is in C Then F(x ) Sup{ F(: x [,] \ C, x < x} = F(x ) ( x c) x is in C and x is in [,] \ C Then x is in some interval Ik and therefore for all x in [,] \ C such that x < x we have that x < x < x and x is in some interval I j at the left of I k and thus, by construction, F ( < F( for all x in [,] \ C with x < x But this implies that Sup{ F(: x [,] \ C, x < x} F(x ), ie, F(x ) F(x ) Thus, in any case, F is monotonically increasing In fact F is strictly increasing, since in cases (a) and (b) above, if F(x ) = F(x ), then this value is in some interval J k, but F is injective there which implies that x = x ; but this contradicts that x < x Property (i) F is continuous and bijective Let us show that F is continuous Since F is monotonically increasing on [,], then by a well known theorem of real analysis (see for example Rudin s Principle of Mathematical Analysis, Theorem 49, p95) we have that for any x (,), + F( x ) : lim F( = inf{ F( : x [,] \ C, x > x} = + x x = lim x x F( x ) : F( = sup{ F( : x [,] \ C, x < x}
3 We claim that sup{ F( : x [,] \ C, x < x} = inf{ F ( : x [,] \ C, x > x} Notice that if for any x < x and any t > x, such that x, t [,] \ C, we have F( F( t) and F sup{ F( : x [,] \ C, x < } ( x which implies that sup{ F( : x [,] \ C, x < x} F( t) for any t > x and therefore sup{ F( : x [,] \ C, x < x } inf{ F( t) : t [,] \ C, t > x = inf{ F( : x [,] \ C, x > x For the reverse inequality, suppose that sup{ F ( : x < x} < inf{ F( : x > x} Since [,] \ C is dense in [,] then any subinterval of [,] contains points of [,] \ C, ie, values of F in [,] \ C Then the subinterval (sup{ F ( : x < x}, inf{ F( : x > x}) must contain values of F Let y = F( x ) be one of such values If x >, then y F x ) inf{ F( : x > }, a contradiction If x = ( x x < x F( sup{ F( : x x}, again a contradiction y = < Thus we conclude that sup{ F( : x [,] \ C, x < x} = inf{ F ( : x [,] \ C, x > x} ie, lim F( = lim F( = F( x) x x x x + This proves that F is continuous at x } }, then By a similar argument we can prove that F is continuous at and at Therefore F is continuous on [,] Let us now show that F is bijective Since F is increasing and continuous on [,] then F is injective For the surjectivity, take any y in [,] Since f is continuous on [,], then by the intermediate value theorem there is an x in [,] such that F( = y Hence F maps [,] onto [,] and therefore F: [,] [,] is bijective
4 Property (iii) F maps surjectively onto C C Notice that:  by construction F: [,] \ C [,] \ C is bijective  by property (i), F: [,] [,] is bijective Therefore F: C C is bijective, ie, F maps C surjectively onto C Remark Since F is a continuous oneone mapping of the compact metric space [,] onto the metric space [,], then the inverse mapping F defined by F ( F( ) = x is a continuous mapping (see for example Rudin s Principles of Mathematical Analysis, Theorem 47, p9) This implies that any two Cantor sets are homeomorphic 35 Give an example of a measurable function f and a continuous function Φ so that f o Φ is nonmeasurable [Hint: Let Φ : C C as in Exercise 34, with m( C ) > and m( C ) = Let N C be nonmeasurable, and take f = χ Φ(N ) ] Use the construction in the hint to show that there exists a Lebesgue measurable set that is not a Borel set Proof Following the hint, let Φ : C C as in Exercise 34, with m( C ) > and m( C ) = Since ( C ) >, there exists a nonmeasurable set N C Now take f m =, x Φ( N) χ = Φ ( N (, x Φ( N ) ( ) Since Φ( N) C ( by 34(iii) ) and m ( C ) =, then m ( Φ( N)) =, so Φ(N) is measurable and therefore f is measurable Now consider f o Φ Notice that f Φ ( o ) ((, + )) = N is nonmeasurable and therefore f o Φ is nonmeasurable ** f Φ ( o ) ((, + )) = N Indeed: ** x N Φ( Φ( N) x ( f o Φ) f ( Φ( ) = (, + ) x ( f ((, + )) ( f o Φ)( (, + ) o Φ) f ( Φ( ) (, + ) f ( Φ( ) = Φ( Φ( N) x Φ ((, + )) ( Φ( N)) = N
5 For the second statement, let Φ : C C as in Exercise 34, with m( C ) = and m( C ) > Since m( C ) >, there exists a nonmeasurable set V C Then A = Φ ( V ) C and since m( C ) = then m( A) = and therefore A is measurable Now suppose that A is a Borel set Then Φ( A ) = Φ( Φ ( V )) = V is also a Borel set since Φ is a continuous oneone function and continuous oneone functions map Borel sets onto Borel sets (see proof of this below) But if V is a Borel set, then V is measurable, a contradiction ** Theorem If f is a oneone continuous mapping of R onto R, then f maps Borel sets onto Borel sets Proof Recall that if f is a oneone map of X onto Y, then for A, B X, f ( A B) = f ( A) f ( B) and f ( A \ B) = f ( A) \ f ( B) Now set U ={ A R : f ( A) B}, where B denotes the collection of Borel sets We claim that U is a σalgebra Indeed, if A U, then A c c U, because f ( A ) = f ( R \ A) = R \ f ( A) Also, if A } is a sequence of U U n= n A n= n U n = sets in U, then f ( A ) = f ( ), which shows that A U Now since f is strictly monotonic (because it is oneone and continuous), we have that f([a,b]) = [f(a),f(b)] or f([a,b]) = [f(b),f(a)] Hence U contains all closed intervals and therefore all Borel sets n { n
6 Exercise 36 This exercise provides an example of a measurable function f on [, ] such that every function g equivalent to f (in the sense that f and g differ only on a set of measure zero) is discontinuous at every point (a) Construct a measurable set E [, ] such that for any nonempty open subinterval I in [, ], both sets E I and E c I have positive measure (b) Show that f = χ E has the property that whenever g( = f( ae x, then g must be discontinuous at every point in [, ] Solution to part (a) Let {u k } k= be a sequence of all intervals with rational endpoints We will denote u = (α, β ), where α, β are rational numbers in [, ] Because u is an open interval, we can construct a Cantorlike set A of positive measure entirely inside u Since Cantor (and Cantorlike) sets are nowhere dense, we can always find an open interval (a, b) inside u, such that (a, b) is disjoint from A Therefore we can construct another Cantorlike set A, such that A (a, b) u Now let s take an open interval u = (α, β ) We can construct Cantorlike set A 3 entirely inside u such that A 3 is disjoint from all previosly constructed sets and, by the same argument we construct A 4 inside u such that it is disjoint from all previous Cantorlike sets Notice that sets A k and A j satisfy the following conditions: (i)a k and A j are disjoint and nowhere dense, (ii)a k u k and A k u k Repeating this process indefinitely we get a sequence A k that has the properties: (i) A k and A k are disjoint and nowhere dense k N, (ii) A k u k and A k u k Let s take E = A k then I k : u k I k= E I = ( A k ) I A k I = Ak () k= To show the second inequality let s notice that m(e I) m(a k ) > () A k is not a subset of E k=
7 This means that A k E c k= E c I ( A k ) I A k I = Ak (3) k= This concludes the proof of part (a) Solution to part (b) m(e c I) m(a k ) > (4) To show that the claim is valid, we will need to use the following Theorem(without proof): The Baire Cathegory Theorem: Let X be a complete metric space, (a) If {U n } is a sequence of open dense subsets of X, then U n is dense in X, (b) X is not a countable union of nowhere dense sets Let f( = χ E and let g( = f( almost everywhere By definition of continuity g( is continuous if and only if: lim g( = g( (5) x x Without loss of generality, let s assume that x I, where I is an open interval in [, ] Let F be the set where f( is different from g(, so that m(f ) = We see that x has no other option, but to satisfy one of the following: () x (E I F c ) () x (E c I F c ) (3) x F As we already proved m(e I) > and m(e c I) > Using the result of part (a) we know that m(e I) > and m(e c I) > therefore g( = f( on E I F c and E c I F c
8 First, let s show that if case () is true, then g is discontinuos: Suppose x E F c Because E c = ( A k ) c = A c k, where Ac k are dense in [, ], according to Baire Cathegory Theorem, E c is dense in [, ] and E c F c is also dense in [, ] Because E c F c is dense we can find a sequence {x n } E c F c converging to x, then g(x n ) = f(x n ) = χ E (x n ), but g( = f( = χ E ( = This implies discontinuity of g Now, let s show that if case () is true, then g is discontinuos, too: Suppose x (E c I F c ) Then g( = f( = χ E ( Now let s find a sequence {x n } n= convergent to x, such that g(x n ) does not converge to g( Among the sequence of intervals with rational endpoints {u k }, let s choose a subsequence of shrinking, nested intervals {u kj }, so that: α kj < α kj+ x β kj+ < β kj, (6) this can be done, because as we know that, any number can be approximated arbitrarily close by a rational number Constructing Cantorlike sets A kj in each of the open intervals {u kj } as suggested in part (a), and by choosing a sequence {x kj } j A kj u kj, we obtain a sequence, converging to x, that is entirely inside E F c Note g(x kj ) = χ E ( j and so g(x kj ) does not converge to g(, this implies that g is discontinuous Therefore g( = f( ae x is discontinuous almost everywhere Using case () and case () we notice for case (3) the following fact: Suppose x F, then in any neiborhood of x we have points from both E and E c and thus, the sequence constructed by taking points from each of the sets E and E c such that x n E and x n E c we construct a sequence convergent to x, that has no limit This contradicts the definition of continuity The conclusion for these three cases gives us the result that g has to be discontinuous everywhere 3
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 informationMetric 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 informationFIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper.
FIRST YEAR CALCULUS WWLCHENW L c WWWL W L Chen, 1982, 2008. 2006. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It It is is
More informationMath212a1010 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 informationRi and. i=1. S i N. and. R R i
The subset R of R n is a closed rectangle if there are n nonempty 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 information1. 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 MenGen 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 informationBasic 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 informationMASSACHUSETTS 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 informationSOLUTIONS 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 informationCHAPTER 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 informationPractice 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
More informationMA4001 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
More information9 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 informationMA651 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 informationMathematics 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 informationThe 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 informationNotes 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 informationChapter 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 informationNo: 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 informationx 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 informationExtension 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.
More informationBANACH 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
More informationand 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 641643, UAB 19992000 M. Griesemer Throughout these notes m denotes Lebesgue measure. 1. Abstract Integration σalgebras. A σalgebra in X is a nonempty collection of subsets
More informationCHAPTER 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 informationIntroduction 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 informationCardinality. 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 informationREAL 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 informationFUNCTIONAL 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 informationThe HenstockKurzweilStieltjes type integral for real functions on a fractal subset of the real line
The HenstockKurzweilStieltjes 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 information1 The Concept of a Mapping
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 1 The Concept of a Mapping The concept of a mapping (aka function) is important throughout mathematics. We have been dealing
More informationLecture 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 informationDiscrete 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
More informationTHE 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,
More informationUndergraduate 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
More informationRecursion 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
More informationMTH4100 Calculus I. Lecture notes for Week 8. Thomas Calculus, Sections 4.1 to 4.4. Rainer Klages
MTH4100 Calculus I Lecture notes for Week 8 Thomas Calculus, Sections 4.1 to 4.4 Rainer Klages School of Mathematical Sciences Queen Mary University of London Autumn 2009 Theorem 1 (First Derivative Theorem
More informationMathematical 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 informationRANDOM INTERVAL HOMEOMORPHISMS. MICHA L MISIUREWICZ Indiana University Purdue University Indianapolis
RANDOM INTERVAL HOMEOMORPHISMS MICHA L MISIUREWICZ Indiana University Purdue University Indianapolis This is a joint work with Lluís Alsedà Motivation: A talk by Yulij Ilyashenko. Two interval maps, applied
More informationThe 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 informationINTRODUCTORY SET THEORY
M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H1088 Budapest, Múzeum krt. 68. CONTENTS 1. SETS Set, equal sets, subset,
More informationSeparation Properties for Locally Convex Cones
Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam
More informationFollow 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 informationMeasure Theory. Jesus FernandezVillaverde University of Pennsylvania
Measure Theory Jesus FernandezVillaverde University of Pennsylvania 1 Why Bother with Measure Theory? Kolmogorov (1933). Foundation of modern probability. Deals easily with: 1. Continuous versus discrete
More informationOur 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 informationPART 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
More informationUltraproducts 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 informationUniversity 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 informationSOLUTIONS 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 informationCHAPTER 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 informationMetric 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{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 informationINDISTINGUISHABILITY 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 informationMEASURE 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 informationComments on Quotient Spaces and Quotient Maps
22M:132 Fall 07 J. Simon Comments on Quotient Spaces and Quotient Maps There are many situations in topology where we build a topological space by starting with some (often simpler) space[s] and doing
More informationGENERIC 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 informationx 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
More informationReading material on the limit set of a Fuchsian group
Reading material on the limit set of a Fuchsian group Recommended texts Many books on hyperbolic geometry and Kleinian and Fuchsian groups contain material about limit sets. The presentation given here
More informationMath 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 informationDISINTEGRATION OF MEASURES
DISINTEGRTION OF MESURES BEN HES Definition 1. Let (, M, λ), (, N, µ) be sigmafinite measure spaces and let T : be a measurable map. (T, µ)disintegration is a collection {λ y } y of measures on M such
More informationOpen 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.
More informationLECTURE 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 eightweek
More information3. Mathematical Induction
3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)
More informationLecture 16 : Relations and Functions DRAFT
CS/Math 240: Introduction to Discrete Mathematics 3/29/2011 Lecture 16 : Relations and Functions Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT In Lecture 3, we described a correspondence
More informationI. 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 informationConnectivity and cuts
Math 104, Graph Theory February 19, 2013 Measure of connectivity How connected are each of these graphs? > increasing connectivity > I G 1 is a tree, so it is a connected graph w/minimum # of edges. Every
More informationRolle s Theorem. q( x) = 1
Lecture 1 :The Mean Value Theorem We know that constant functions have derivative zero. Is it possible for a more complicated function to have derivative zero? In this section we will answer this question
More informationLebesgue Measure on R n
8 CHAPTER 2 Lebesgue Measure on R n Our goal is to construct a notion of the volume, or Lebesgue measure, of rather general subsets of R n that reduces to the usual volume of elementary geometrical sets
More informationON SEQUENTIAL CONTINUITY OF COMPOSITION MAPPING. 0. Introduction
ON SEQUENTIAL CONTINUITY OF COMPOSITION MAPPING Abstract. In [1] there was proved a theorem concerning the continuity of the composition mapping, and there was announced a theorem on sequential continuity
More informationMathematics for Computer Science
Mathematics for Computer Science Lecture 2: Functions and equinumerous sets Areces, Blackburn and Figueira TALARIS team INRIA Nancy Grand Est Contact: patrick.blackburn@loria.fr Course website: http://www.loria.fr/~blackbur/courses/math
More informationn 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 informationGROUPS 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 informationWe 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 informationSo 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
More informationSet theory as a foundation for mathematics
Set theory as a foundation for mathematics Waffle Mathcamp 2011 In school we are taught about numbers, but we never learn what numbers really are. We learn rules of arithmetic, but we never learn why these
More informationINTRODUCTION TO DESCRIPTIVE SET THEORY
INTRODUCTION TO DESCRIPTIVE SET THEORY ANUSH TSERUNYAN Mathematicians in the early 20 th century discovered that the Axiom of Choice implied the existence of pathological subsets of the real line lacking
More informationFixed Point Theorems
Fixed Point Theorems Definition: Let X be a set and let T : X X be a function that maps X into itself. (Such a function is often called an operator, a transformation, or a transform on X, and the notation
More informationTHE PRIME NUMBER THEOREM
THE PRIME NUMBER THEOREM NIKOLAOS PATTAKOS. introduction In number theory, this Theorem describes the asymptotic distribution of the prime numbers. The Prime Number Theorem gives a general description
More informationFinite 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
More informationMathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson
Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement
More informationContinuity. DEFINITION 1: A function f is continuous at a number a if. lim
Continuity DEFINITION : A function f is continuous at a number a if f(x) = f(a) REMARK: It follows from the definition that f is continuous at a if and only if. f(a) is defined. 2. f(x) and +f(x) exist.
More informationON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE. 1. Introduction
ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE J.M. CALABUIG, J. RODRÍGUEZ, AND E.A. SÁNCHEZPÉREZ Abstract. Let m be a vector measure taking values in a Banach space X. We prove that
More informationA 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
More informationThe Mean Value Theorem
The Mean Value Theorem THEOREM (The Extreme Value Theorem): If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers
More informationGod created the integers and the rest is the work of man. (Leopold Kronecker, in an afterdinner 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 afterdinner speech at a conference, Berlin, 1886) God created the integers and the rest is the work
More informationPoint 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 informationHomework # 3 Solutions
Homework # 3 Solutions February, 200 Solution (2.3.5). Noting that and ( + 3 x) x 8 = + 3 x) by Equation (2.3.) x 8 x 8 = + 3 8 by Equations (2.3.7) and (2.3.0) =3 x 8 6x2 + x 3 ) = 2 + 6x 2 + x 3 x 8
More informationChapter 1  σalgebras
Page 1 of 17 PROBABILITY 3  Lecture Notes Chapter 1  σalgebras 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 informationDegrees that are not degrees of categoricity
Degrees that are not degrees of categoricity Bernard A. Anderson Department of Mathematics and Physical Sciences Gordon State College banderson@gordonstate.edu www.gordonstate.edu/faculty/banderson Barbara
More informationThis 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 information36 CHAPTER 1. LIMITS AND CONTINUITY. Figure 1.17: At which points is f not continuous?
36 CHAPTER 1. LIMITS AND CONTINUITY 1.3 Continuity Before Calculus became clearly de ned, continuity meant that one could draw the graph of a function without having to lift the pen and pencil. While this
More informationMATH 110 Spring 2015 Homework 6 Solutions
MATH 110 Spring 2015 Homework 6 Solutions Section 2.6 2.6.4 Let α denote the standard basis for V = R 3. Let α = {e 1, e 2, e 3 } denote the dual basis of α for V. We would first like to show that β =
More informationFoundations of Mathematics I Set Theory (only a draft)
Foundations of Mathematics I Set Theory (only a draft) Ali Nesin Mathematics Department Istanbul Bilgi University Kuştepe Şişli Istanbul Turkey anesin@bilgi.edu.tr February 12, 2004 2 Contents I Naive
More informationMetric 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
More information1. 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. 20050314 Skrivtid: 9 11.30 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
More informationON FIBER DIAMETERS OF CONTINUOUS MAPS
ON FIBER DIAMETERS OF CONTINUOUS MAPS PETER S. LANDWEBER, EMANUEL A. LAZAR, AND NEEL PATEL Abstract. We present a surprisingly short proof that for any continuous map f : R n R m, if n > m, then there
More informationDEGREES OF ORDERS ON TORSIONFREE ABELIAN GROUPS
DEGREES OF ORDERS ON TORSIONFREE ABELIAN GROUPS ASHER M. KACH, KAREN LANGE, AND REED SOLOMON Abstract. We construct two computable presentations of computable torsionfree abelian groups, one of isomorphism
More informationBasic 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
More informationReal Roots of Univariate Polynomials with Real Coefficients
Real Roots of Univariate Polynomials with Real Coefficients mostly written by Christina Hewitt March 22, 2012 1 Introduction Polynomial equations are used throughout mathematics. When solving polynomials
More informationA CONSTRUCTION OF THE UNIVERSAL COVER AS A FIBER BUNDLE
A CONSTRUCTION OF THE UNIVERSAL COVER AS A FIBER BUNDLE DANIEL A. RAMRAS In these notes we present a construction of the universal cover of a path connected, locally path connected, and semilocally simply
More informationMatrix Representations of Linear Transformations and Changes of Coordinates
Matrix Representations of Linear Transformations and Changes of Coordinates 01 Subspaces and Bases 011 Definitions A subspace V of R n is a subset of R n that contains the zero element and is closed under
More information