Our goal first will be to define a product measure on A 1 A 2.
|
|
- Maximillian Russell
- 7 years ago
- Views:
Transcription
1 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 direct products of elements of A 1,2. That is, A 1 A 2 = σ Ω1 ({E 1 E 2 : E 1,2 A 1,2 }). Our goal first will be to define a product measure on A 1 A A measure ν defined on the σ -algebra A 1 A 2, such that ν(e 1 E 2 ) = µ 1 (E 1 )µ 2 (E 2 ) E 1,2 A 1,2 is called the (tensor) product of µ 1 and µ 2. A somewhat surprising fact is that in such general situation the product ν is not unique, although always exists. We will denote any product measure by µ 1 µ 2. Thus ν = µ 1 µ 2 makes sence, but µ 1 µ 2 = ν is a uniqueness statement (which might be wrong). 3. We shall establish the existence of µ 1 µ 2 by running the Caratheodory maschine. We need to prepare the setting for it. In the following notice the strong similarities with the construction of λ n on R n. 4. A (measurable) rectangle is the set of the form X Y with X A 1, Y A 2. Notice that the intersection of rectangles is a rectangle (A B) (X Y ) = (A X) (B Y ), and that the complement of a rectangle is the disjoint union of two rectangles (A B) c = (A c ) (A B c ). These observations will be useful later. 5. For a rectangle X Y define π(x Y ) = µ 1 (X)µ 2 (Y ). The following lemma will be the main tool in the later proofs. Lemma 1 Let A, A 1,2... A 1, B, B 1,2,... A 2 are disjoint, and A B = A j B j. j be such that all A j B j Then π(a B) = j π(a j B j ). 1
2 Let rectangles C 1 D 1, C 2 D 2,..., C j A 1, D j A 2, cover A B, A B j C j D j. Then π(a B) j π(c j D j ). Proof. 1. For both parts of the theorem the key is the following observation. In the first case the formula χ A B (x, y) = χ A (x)χ B (y) = j χ Aj (x)χ Bj (y) holds for all (x, y). In the second case χ A B (x, y) = χ A (x)χ B (y) j χ Aj (x)χ Bj (y). 2. Let us prove, for example, the case of equality in the lemma. The other case is proved the same way. Fix x and think of the functions in the formula as maps from. Apply the monotone convergence theorem on the measure space (A 2,, µ 2 ). Deduce that χ A (x)µ 2 (B) = χ A (x) χ B (y) dµ 2 (y) = χ Aj (x) χ Bj (y) dµ 2 (y) j = j χ Aj (x)µ 2 (B j ). Next, think of the last formula as the identity for maps from. Application of the monotone convergence on (A 1,, µ 1 ) completes the proof. 6. For any E define the outer measure π (E) = inf j π(a j B j ), where the infimum is taken over all coverings of E by at most countable number of measurable rectangles A j B j. Similarly to the construction of the Lebesgue measure in R n we establish (by a different proof) the following proposition. 2
3 Proposition 2 The outer measure π : 2 Ω1 Ω2 following properties: (i) π ( ) = 0 ; (ii) π (E 1 ) π (E 2 ) if E 1 E 2 ; (iii) semiadditivity holds, that is [0, + ] enjoys the π j E j j π (E j ); (iv) π (A B) = π(a B) for any measurable rectangle A B. Proof. 1. The proofs of (i) (iii) are exactly the same as in the corresponding proposition in part 1. However (iv) requires a completely different approach than in part 1 (why?). 2. On the one hand by the definition π (A B) π(a B). On the other hand for any disjoint cover of A B by rectangles {A j B j } we have A B (A B) A j B j j = j (A A j ) (B B j ). Hence by Lemma 1 π(a B) j π((a A j ) (B B j )) j π(a j B j ). We conclude the proof of (iv) by taking the infimum over all covers in the latter formula. 7. We see that π satisfies all axioms of the outer measure in the Caratheodory construction which we developed in part 1. As we emphasized there, the theory relies on nothing except the axioms. Apply the Caratheodory construction to the outer measure π and obtain a σ -algebra (P, ) of π -measurable sets defined by { P = E : π (X) = π (X E) +π (X E c ) X }. 3
4 Moreover, according to the Caratheodory construction, the restriction π P is a complete measure on P. 8. Theorem 3 Let (A j, Ω j, µ j ), j = 1, 2, be two measure spaces. Then any rectangle A B, A A 1, B A 2, is π -measurable, and therefore A 1 A 2 P. Moreover π (A B) = µ 1 (A)µ 2 (B). Proof. 1. The equality was proved in Proposition 2 (iv). We just need to show that any rectangle is π -measurable. 2. Set S = A B. To show the measurability of S we take any test set X. Fix any ε > 0. Cover this X by a family of rectangles {R j } so that π(r j ) π (X) + ε. For each j we write j R j = (R j S) (R j S c ), where the union is disjoint. Every R j S is a rectangle and X S is covered by {R j S}. Similarly R j S c is a disjoint union of at most two rectangles, R j S c = C 1j C 2j, and X S c is covered by {C 1j, C 2j }. For every j by Lemma 1 Therefore π(r j ) = π(r j S) + π(c 1j ) + π(c 2j ). π (X) + ε j π(r j S) + π(c 1j ) + π(c 2j ) π (X S) + π (X S c ). We can take ε arbitrary small. Hence S is measurable. 9. For any pair of the measure spaces (A j, Ω j, µ j ), j = 1, 2, we constructed π = µ 1 µ 2 defined on A 1 A 2. In general situation the tensor product of measures is not unique (cf. below). However, if the terms µ 1,2 are σ - finite, then µ 1 µ 2 is unique as the following theorem states. 10. On the σ -algebra (R, 2 R ) consider the measure µ, { 0, E is at most countable µ(e) =, E is uncountable. Set A = 2 R 2 R. Prove the following statements about µ µ. 4
5 (a) For E A define ν(e) = 0, if E = G H for some G, H A, such that proj x (G) and proj y (H) are both countable. Otherwise define ν(e) =. Prove that ν is a measure on A, and that ν = µ µ. (b) Let L = {(x, y): x = y} be the bisector line in R 2. For E A define ρ(e) = 0, if E = G H K for some G, H, K A, such that proj x (G), proj y (H), and proj L (K) are all countable. Otherwise define ρ(e) =. Prove that ρ is a measure on A, and that ρ = µ µ. (c) It will be helpful to prove that if ν(e) = 0 or ρ(e) = 0, then E is contained in a countable set of lines. (d) Define E 0 = {(x, y): x = y}. Prove that E 0 A, and ν(e 0 ) ρ(e 0 ). 11. We shall prove that any µ 1 µ 2 must be equal to π obtained from the Caratheodory construction, provided the terms are σ -finite. Theorem 4 Let (A j, Ω j, µ j ), j = 1, 2, be two measure spaces with σ - finite µ 1,2. Let ν be any measure on A 1 A 2 such that ν(x 1 X 2 ) = µ 1 (X 1 )µ 2 (X 2 ) for all X 1,2 A 1,2. Then ν = π on A 1 A 2. Proof. 1. First suppose µ 1,2 <. Then both ν and π are also finite. Take any E A 1 A 2. For any sequence of rectangles {R j } covering E we derive ν(e) ν j R j j ν(r j ) = π(r j ). Take the infimum over all such coverings to discover On the other hand in the formula ν(e) π (E) E A 1 A 2. (0.1) ν(e) + ν(e c ) = µ 1 ( )µ 2 ( ) = π (E) + π (E c ) we utilise the finiteness of all terms to deduce ν(e) = π (E) + π (E c ) ν(e c ) π (E) since π (E c ) ν(e c ) 0 by (0.1). 2. Write down a limit argument from the finite to the σ -finite case to complete the proof of the theorem. 5
6 12. Now we turn to the proerties of the integral with respect to µ 1 µ 2. Our goal is to establish relation between the "double" and "repeated" integrals. This circle of ideas is usually called the Fubini s theorem, despite there are more than one theorem there. We need to introduce the following objects. For any E define its sections by writing and E x def = {y : (x, y) E}, E y def = {x : (x, y) E}, Similarly for a function f : R define its sections by and Thus f x = f(x, ), f y = f(, y). f x : R y f(x, y), f y : R x f(x, y). 13. Lemma 5 (i) For any E A 1 A 2 and any x, y the sections are measurable: E x A 2, E y A 1. (ii) For any A 1 A 2 -measurable function f the sections are measurable: f x is A 2 -measurable, f y is A 1 -measurable. Proof. 1. Prove (i) using the principle of good sets from the part 1. Namely, prove that the collection of all sets E, such that all E x, E y are measurable, is a σ -algebra. Then prove that all rectangles are in this collection. Conclude then that the entire A 1 A 2 is also a part of this collection. 2. Fix any x Let us show, for example, that f 1 x ((t, + )) A 2 for any t R. Indeed, clearly fx 1 ((t, + )) = {y : f(x, y) > t} = ({(x, y) : f(x, y) > t}) x. Hence the statement follows from the measurability of f and part (i). 14. The main technical part of the Fubini s theory is the following lemma. Lemma 6 Let (A j, Ω j, µ j ), j = 1, 2, be two measure spaces with σ - finite µ 1,2. For any E A 1 A 2 define f 1,2 :,2 R by writing f 1 (x) = µ 2 (E x ), x f 2 (y) = µ 1 (E y ), y. 6
7 Then f 1,2 are A 1,2 -measurable, and µ 1 µ 2 (E) = f 1 dµ 1 = f 2 dµ Accepting this lemma let us prove the main theorems. The following Tonelli s theorem claims that for positive measurable functions the finiteness of the "double" integral is equivalent to the finiteness of the "repeated". We will see later that for sign changing functions this is not the case. Theorem 7 Let (A j, Ω j, µ j ), j = 1, 2, be two measure spaces with σ - finite µ 1,2. Let F : R, F 0, be A 1 A 2 -measurable. Then the functions f(x) = F x dµ 2, x, and g(y) = F y dµ 1, y, are measurable, and f dµ 1 = g dµ 2 = F d(µ 1 µ 2 ). In other symbols, ( ) F (x, y) dµ 2 (y) dµ 1 (x) = = ( ) F (x, y) dµ 1 (x) dµ 2 (y) F (x, y) d(µ 1 µ 2 )(x, y). In this theorem we require only that F is A 1 A 2 -measurable. The integrals can be finite or infinite. 16. Proof of Theorem By Lemma 6 the theorem holds for the case F = χ E, E A 1 A 2. Then by linearity we at once conclude that the theorem holds for any simple F. The general case will follow by a careful approximation procedure. 2. Fix any nonnegative measurable F. By the approximation theorem there exists a monotone sequence of positive simple functions {Φ n }, Φ n Φ n+1 on, 7
8 such that Φ n F as n, pointwisely on. For each n define φ n (x) = (Φ n ) x dµ 2, x. From Lemma 6 it follows that for any n the function φ n is A 1 - measurable and Φ n d(µ 1 µ 2 ) = φ n dµ 1. (0.2) We would like to pass in (0.2) to the limit as n. The left hand side there is easy: the monotone convergence gives at once that Φ n d(µ 1 µ 2 ) F d(µ 1 µ 2 ), n. In order to pass to the limit in the right hand side of (0.2) notice first that for any fixed x (Φ n )x (Φ n+1)x on, and that (Φ n ) x F x, n, on. All functions here are measurable as the sections of measurable functions. Hence by the monotone convergence we discover that for any fixed x φ n (x) = (Φ n ) x dµ 2 Ω 2 F x dµ 2, as n. Monotonicity of {Φ n } implies that φ n φ n+1 on. Once more apply the monotone convergence on (, A 1, µ 1 ) to derive ( ) φ n (x) dµ 1 (x) F x dµ 2 dµ 1 (x), as n. Thus passing to the limit as n in (0.2) we deduce ( ) F d(µ 1 µ 2 ) = F x dµ 2 dµ 1 (x) = f dµ The proof of F d(µ 1 µ 2 ) = proceeds verbatim. ( ) F y dµ 1 dµ 2 (y) = g dµ 2 8
9 17. The following theorem due to Fubini covers the case of sign changing functions. Theorem 8 Let (A j, Ω j, µ j ), j = 1, 2, be two measure spaces with σ - finite µ 1,2. Suppose F d(µ 1 µ 2 ) < ( that is F L 1 (, µ 1 µ 2 ) ). Define the functions f(x) = F x dµ 2, x, and g(y) = F y dµ 1, y. Then f L 1 (, µ 1 ), g L 1 (, µ 2 ), and f dµ 1 = g dµ 2 = F d(µ 1 µ 2 ). In other symbols, ( ) F (x, y) dµ 2 (y) dµ 1 (x) = = ( ) F (x, y) dµ 1 (x) dµ 2 (y) Ω 1 F (x, y) d(µ 1 µ 2 )(x, y). Proof. Prove the theorem. Hint: split F into a positive and negative part, F = F + F, and derive the statement from Tonelli s theorem. Indicate clearly the step when the assumption F L 1 is crucial. 18. Give an example showing that the Fubini s theorem fails if F is only A 1 A 2 -measurable. 9
Lecture Notes on Measure Theory and Functional Analysis
Lecture Notes on Measure Theory and Functional Analysis P. Cannarsa & T. D Aprile Dipartimento di Matematica Università di Roma Tor Vergata cannarsa@mat.uniroma2.it daprile@mat.uniroma2.it aa 2006/07 Contents
More informationHow To Find Out How To Calculate A Premeasure On A Set Of Two-Dimensional Algebra
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 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 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 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 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 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 Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since
More 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 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 641-643, UAB 1999-2000 M. Griesemer Throughout these notes m denotes Lebesgue measure. 1. Abstract Integration σ-algebras. A σ-algebra in X is a non-empty collection of subsets
More 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 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 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 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 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 informationI. 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.
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 informationSample Induction Proofs
Math 3 Worksheet: Induction Proofs III, Sample Proofs A.J. Hildebrand Sample Induction Proofs Below are model solutions to some of the practice problems on the induction worksheets. The solutions given
More information1 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 informationDISINTEGRATION OF MEASURES
DISINTEGRTION OF MESURES BEN HES Definition 1. Let (, M, λ), (, N, µ) be sigma-finite measure spaces and let T : be a measurable map. (T, µ)-disintegration is a collection {λ y } y of measures on M such
More informationIEOR 6711: Stochastic Models I Fall 2012, Professor Whitt, Tuesday, September 11 Normal Approximations and the Central Limit Theorem
IEOR 6711: Stochastic Models I Fall 2012, Professor Whitt, Tuesday, September 11 Normal Approximations and the Central Limit Theorem Time on my hands: Coin tosses. Problem Formulation: Suppose that I have
More informationMarkov random fields and Gibbs measures
Chapter Markov random fields and Gibbs measures 1. Conditional independence Suppose X i is a random element of (X i, B i ), for i = 1, 2, 3, with all X i defined on the same probability space (.F, P).
More informationMATH 4330/5330, Fourier Analysis Section 11, The Discrete Fourier Transform
MATH 433/533, Fourier Analysis Section 11, The Discrete Fourier Transform Now, instead of considering functions defined on a continuous domain, like the interval [, 1) or the whole real line R, we wish
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 information6.2 Permutations continued
6.2 Permutations continued Theorem A permutation on a finite set A is either a cycle or can be expressed as a product (composition of disjoint cycles. Proof is by (strong induction on the number, r, of
More informationGod created the integers and the rest is the work of man. (Leopold Kronecker, in an after-dinner speech at a conference, Berlin, 1886)
Chapter 2 Numbers God created the integers and the rest is the work of man. (Leopold Kronecker, in an after-dinner speech at a conference, Berlin, 1886) God created the integers and the rest is the work
More information1 Norms and Vector Spaces
008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)
More 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 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 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 informationProbability Theory. Florian Herzog. A random variable is neither random nor variable. Gian-Carlo Rota, M.I.T..
Probability Theory A random variable is neither random nor variable. Gian-Carlo Rota, M.I.T.. Florian Herzog 2013 Probability space Probability space A probability space W is a unique triple W = {Ω, F,
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column
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 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 informationThe 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
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 informationSolutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
More informationThe sample space for a pair of die rolls is the set. The sample space for a random number between 0 and 1 is the interval [0, 1].
Probability Theory Probability Spaces and Events Consider a random experiment with several possible outcomes. For example, we might roll a pair of dice, flip a coin three times, or choose a random real
More informationHOMEWORK 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
More information(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
More informationON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP. A. K. Das and R. K. Nath
International Electronic Journal of Algebra Volume 7 (2010) 140-151 ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP A. K. Das and R. K. Nath Received: 12 October 2009; Revised: 15 December
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 informationWHERE DOES THE 10% CONDITION COME FROM?
1 WHERE DOES THE 10% CONDITION COME FROM? The text has mentioned The 10% Condition (at least) twice so far: p. 407 Bernoulli trials must be independent. If that assumption is violated, it is still okay
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
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 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 informationa 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
More informationThis asserts two sets are equal iff they have the same elements, that is, a set is determined by its elements.
3. Axioms of Set theory Before presenting the axioms of set theory, we first make a few basic comments about the relevant first order logic. We will give a somewhat more detailed discussion later, but
More informationMathematical Induction. Lecture 10-11
Mathematical Induction Lecture 10-11 Menu Mathematical Induction Strong Induction Recursive Definitions Structural Induction Climbing an Infinite Ladder Suppose we have an infinite ladder: 1. We can reach
More informationDEGREES OF ORDERS ON TORSION-FREE ABELIAN GROUPS
DEGREES OF ORDERS ON TORSION-FREE ABELIAN GROUPS ASHER M. KACH, KAREN LANGE, AND REED SOLOMON Abstract. We construct two computable presentations of computable torsion-free abelian groups, one of isomorphism
More informationMath 319 Problem Set #3 Solution 21 February 2002
Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod
More information(IALC, Chapters 8 and 9) Introduction to Turing s life, Turing machines, universal machines, unsolvable problems.
3130CIT: Theory of Computation Turing machines and undecidability (IALC, Chapters 8 and 9) Introduction to Turing s life, Turing machines, universal machines, unsolvable problems. An undecidable problem
More informationSystems of Linear Equations
Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and
More informationCONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12
CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.
More informationLinear Algebra I. Ronald van Luijk, 2012
Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.
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 informationarxiv:math/0606467v2 [math.co] 5 Jul 2006
A Conjectured Combinatorial Interpretation of the Normalized Irreducible Character Values of the Symmetric Group arxiv:math/0606467v [math.co] 5 Jul 006 Richard P. Stanley Department of Mathematics, Massachusetts
More informationFinite dimensional C -algebras
Finite dimensional C -algebras S. Sundar September 14, 2012 Throughout H, K stand for finite dimensional Hilbert spaces. 1 Spectral theorem for self-adjoint opertors Let A B(H) and let {ξ 1, ξ 2,, ξ n
More informationLemma 5.2. Let S be a set. (1) Let f and g be two permutations of S. Then the composition of f and g is a permutation of S.
Definition 51 Let S be a set bijection f : S S 5 Permutation groups A permutation of S is simply a Lemma 52 Let S be a set (1) Let f and g be two permutations of S Then the composition of f and g is a
More informationv w is orthogonal to both v and w. the three vectors v, w and v w form a right-handed set of vectors.
3. Cross product Definition 3.1. Let v and w be two vectors in R 3. The cross product of v and w, denoted v w, is the vector defined as follows: the length of v w is the area of the parallelogram with
More informationChapter 3. Cartesian Products and Relations. 3.1 Cartesian Products
Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing
More informationNormal distribution. ) 2 /2σ. 2π σ
Normal distribution The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a
More informationGalois representations with open image
Galois representations with open image Ralph Greenberg University of Washington Seattle, Washington, USA May 7th, 2011 Introduction This talk will be about representations of the absolute Galois group
More informationF. 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
More informationALMOST 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
More information6.3 Conditional Probability and Independence
222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted
More informationMath 4310 Handout - Quotient Vector Spaces
Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable
More informationBasic Probability Concepts
page 1 Chapter 1 Basic Probability Concepts 1.1 Sample and Event Spaces 1.1.1 Sample Space A probabilistic (or statistical) experiment has the following characteristics: (a) the set of all possible outcomes
More information1. Let P be the space of all polynomials (of one real variable and with real coefficients) with the norm
Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 005-06-15 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
More information1 Definition of a Turing machine
Introduction to Algorithms Notes on Turing Machines CS 4820, Spring 2012 April 2-16, 2012 1 Definition of a Turing machine Turing machines are an abstract model of computation. They provide a precise,
More information1 The Line vs Point Test
6.875 PCP and Hardness of Approximation MIT, Fall 2010 Lecture 5: Low Degree Testing Lecturer: Dana Moshkovitz Scribe: Gregory Minton and Dana Moshkovitz Having seen a probabilistic verifier for linearity
More informationHandout #1: Mathematical Reasoning
Math 101 Rumbos Spring 2010 1 Handout #1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty or
More informationWRITING PROOFS. Christopher Heil Georgia Institute of Technology
WRITING PROOFS Christopher Heil Georgia Institute of Technology A theorem is just a statement of fact A proof of the theorem is a logical explanation of why the theorem is true Many theorems have this
More informationCS 3719 (Theory of Computation and Algorithms) Lecture 4
CS 3719 (Theory of Computation and Algorithms) Lecture 4 Antonina Kolokolova January 18, 2012 1 Undecidable languages 1.1 Church-Turing thesis Let s recap how it all started. In 1990, Hilbert stated a
More informationMATH 425, PRACTICE FINAL EXAM SOLUTIONS.
MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationElements of probability theory
2 Elements of probability theory Probability theory provides mathematical models for random phenomena, that is, phenomena which under repeated observations yield di erent outcomes that cannot be predicted
More informationTiers, Preference Similarity, and the Limits on Stable Partners
Tiers, Preference Similarity, and the Limits on Stable Partners KANDORI, Michihiro, KOJIMA, Fuhito, and YASUDA, Yosuke February 7, 2010 Preliminary and incomplete. Do not circulate. Abstract We consider
More informationThe sum of digits of polynomial values in arithmetic progressions
The sum of digits of polynomial values in arithmetic progressions Thomas Stoll Institut de Mathématiques de Luminy, Université de la Méditerranée, 13288 Marseille Cedex 9, France E-mail: stoll@iml.univ-mrs.fr
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 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 informationINCIDENCE-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
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 information5.3 Improper Integrals Involving Rational and Exponential Functions
Section 5.3 Improper Integrals Involving Rational and Exponential Functions 99.. 3. 4. dθ +a cos θ =, < a
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. 2005-03-14 Skrivtid: 9 11.30 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
More informationMathematical Induction
Mathematical Induction In logic, we often want to prove that every member of an infinite set has some feature. E.g., we would like to show: N 1 : is a number 1 : has the feature Φ ( x)(n 1 x! 1 x) How
More informationLet H and J be as in the above lemma. The result of the lemma shows that the integral
Let and be as in the above lemma. The result of the lemma shows that the integral ( f(x, y)dy) dx is well defined; we denote it by f(x, y)dydx. By symmetry, also the integral ( f(x, y)dx) dy is well defined;
More informationLEARNING OBJECTIVES FOR THIS CHAPTER
CHAPTER 2 American mathematician Paul Halmos (1916 2006), who in 1942 published the first modern linear algebra book. The title of Halmos s book was the same as the title of this chapter. Finite-Dimensional
More informationChapter 20. Vector Spaces and Bases
Chapter 20. Vector Spaces and Bases In this course, we have proceeded step-by-step through low-dimensional Linear Algebra. We have looked at lines, planes, hyperplanes, and have seen that there is no limit
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 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 informationSet theory as a foundation for mathematics
V I I I : Set theory as a foundation for mathematics This material is basically supplementary, and it was not covered in the course. In the first section we discuss the basic axioms of set theory and the
More information1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two non-zero vectors u and v,
1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It
More informationLecture Note 1 Set and Probability Theory. MIT 14.30 Spring 2006 Herman Bennett
Lecture Note 1 Set and Probability Theory MIT 14.30 Spring 2006 Herman Bennett 1 Set Theory 1.1 Definitions and Theorems 1. Experiment: any action or process whose outcome is subject to uncertainty. 2.
More informationTOPIC 4: DERIVATIVES
TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the
More informationFACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z
FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization
More informationThe Graphical Method: An Example
The Graphical Method: An Example Consider the following linear program: Maximize 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2 0, where, for ease of reference,
More informationOn strong fairness in UNITY
On strong fairness in UNITY H.P.Gumm, D.Zhukov Fachbereich Mathematik und Informatik Philipps Universität Marburg {gumm,shukov}@mathematik.uni-marburg.de Abstract. In [6] Tsay and Bagrodia present a correct
More informationMath/Stats 425 Introduction to Probability. 1. Uncertainty and the axioms of probability
Math/Stats 425 Introduction to Probability 1. Uncertainty and the axioms of probability Processes in the real world are random if outcomes cannot be predicted with certainty. Example: coin tossing, stock
More informationDuality 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
More informationBasic Proof Techniques
Basic Proof Techniques David Ferry dsf43@truman.edu September 13, 010 1 Four Fundamental Proof Techniques When one wishes to prove the statement P Q there are four fundamental approaches. This document
More information