Singular Continuous Measures by Michael Pejic 5/14/10
|
|
- Ethan Wood
- 6 years ago
- Views:
Transcription
1 Sigular Cotiuous Measures by Michael Peic 5/4/0 Prelimiaries Give a set X, a σ-algebra o X is a collectio of subsets of X that cotais X ad ad is closed uder complemetatio ad coutable uios hece, coutable itersectios as well. A measure space is a triplet X, A, µ, where X is a set, A is a σ-algebra o X ad the measure µ : A [0, ] satisfies: i µ = 0; ii if {A } J A is a coutable collectio of disoit sets, the µ A = µ A J J If µ X is fiite, µ is termed a fiite measure; if X ca be expressed as the coutable uio of sets i A, each of which has fiite measure, the µ is termed a σ-fiite measure. The support of a measure is the largest closed set such that ay ope subset of it has ozero measure. A fuctio f : X [, ] is A-measurable if the preimage f c, ] is i A for all c R hece the preimage of ay Borel set is i A. A simple A-measurable fuctio is oe takig o oly fiitely may distict values, all of which are fiite. Such a fuctio is itegrable if its support has fiite measure. The itegral of a positive A-measurable fuctio with respect to measure µ is defied as { } f dµ = sup a µ A itegrable simple fuctios a χ A f poitwise The itegral of a geeral A-measurable fuctio with respect to measure µ is defied by f dµ = f >0 dµ f <0 dµ Note that this excludes ay otio of coditioal covergece. Fially, a property is said to hold almost everywhere with respect to µ if it fails o a set of measure zero. Examples i The trivial measure which assigs measure 0 to every elemet of the σ- algebra.
2 ii As a example of fiite measures, probability measures Ω, E, π with π Ω =. iii As a example of a σ-fiite measure, Lebesgue measure with X = R, A give by Borel sets, ad measure λ give by λ A = if b a all coutable collectios of ope itervals such that a, b A The cotiuous fuctios f : [a, b] R are measurable ad for them the defiitio of itegratio agrees with the stadard Riema itegral, b f x dx = f χ [a,b] dλ = f dλ a Stieltes Itegral For X = R ad A give by Borel sets, there is a equivalet formulatio. Let ϕ : R be odecreasig ad cotiuous from the left. Let the Lebesgue- Stieltes measure µ ϕ be the measure iduced by µ ϕ [a, b = ϕ b ϕ a. Coversely, give ay measure o Borel subsets of R that is fiite o bouded sets, µ, defie ϕ : R R by µ [x, 0 if x < 0 ϕ x = 0 if x = 0 µ [0, x if x > 0 The µ = µ ϕ. For cotiuous fuctios f : [a, b] R, this agrees with the stadard Riema-Stieltes itegral, b f dµ ϕ = f x dϕ x = f x ϕ x+ ϕ x [a,b] a lim max k x k+ x k 0 =0 where the limit is take over all partitios a = x 0 x 0 x... x = b. Examples i For ϕ x = x, we recover the Lebesgue measure, µ ϕ = λ. ii For ϕ = Θ, the Heaviside fuctio { 0 if x 0 Θ x = if x > 0 [a,b]
3 we get a measure µ Θ that has a atom of mass at x = 0, which is typically for those igorat of measure theory deoted dϕ = δ x dx for δ the Dirac delta fuctio. Note this measure, ulike Lebesgue measure, ca be exteded from the σ-algebra of Borel sets to the σ-algebra of all subsets of R. Note that discotiuities i ϕ correspod to atoms with mass give by the height of the ump. Absolute Cotiuity Give two A-measures, µ, ν, if a set is of measure 0 with respect to ν wheever it is measure 0 with respect to µ, the ν is said to be absolutely cotiuous with respect to µ, deoted ν µ. If µ, ν are i additio σ-fiite, by the Rado-Nikodym theorem there is a A-measurable fuctio, deoted dν dµ, such that for ay A A, ν A = A dν dµ dµ This fuctio is uique withi a µ-ull fuctio. Mutual Sigularity Give two A-measures, µ, ν, if there are disoit sets E, F such that for ay set A A, A E, A F A ad µ A E = µ A, ν A F = ν A, the µ, ν are said to be mutually sigular, deoted µ ν. Decompositio of a Measure For two σ-fiite, A-measures µ, ν, by the Lebesgue decompostio theorem µ = µ ac + µ sig, where µ ac ν ad µ sig ν. The the Rado-Nikodym derivative dµ dν is defied to be dµac dν. Example Take λ ad µ Θ. These are mutually sigular sice for ay Borel set A, λ A = λ A\ {0} whereas µ Θ A = µ Θ A {0}. Specializig to the case where oe of the measures is Lebesgue measure, there is a further decompositio of the sigular part ito atoms ad a sigular cotiuous part. The first is familiar from physical cocepts of poit masses or charges; the secod is far less so.
4 First Example of Sigular Cotiuous Measure: Devil s Staircase Cosider the sequece of o-decreasig, cotiuous fuctios f =0 from the uit iterval oto itself with f [, ] = f [ 9, 9] = 4, f [, ] =, f [ 7 9, 8 9] = 4 f [ 7, 7] = 8, f [ 9, 9] = 4, f [ 7 7, 8 7] = 8, f [, ] =, f [ 9 7, 0 7] = 5 8, f [ 7 9, 8 9] = 4, f [ 5 7, 6 7] = 7 8. with iterveig values determied by liear iterpolatio. This coverges i supremum orm to a o-decreasig, cotiuous fuctio f : [0, ] [0, ] that clearly has derivative 0 almost everywhere with respect to Lebesgue measure sice the Lebesgue measure of the middle-third Cator set C is 0, so the Rado-Nikodym derivative dµ f = 0, yet µ dλ f is certaily ot 0 sice µ f [0, ] =. At the same time, sice f is cotiuous, µ f cotais o atoms. Therefore, µ f is purely sigular cotiuous. Note µ f is equal to the Hausdorff measure h log defied by h log A = lim ε 0 + if diam B log B B ope covers B of A by balls of diameter less tha ε Doig itegrals with respect to µ f by takig the limit as of doig them with µ f is tedious. I some cases, this ca be made easier by iterpretig the graph of f as a affie iterated fractal system with base B = 0, 0,, 0, 0, ad geerators G = 0, 0,, 0, 0,, G =,,,,,, ad G =,,,,, with correspodig affie trasformatios A x, y = x, y A x, y = x +, 4
5 A x, y = x +, y + so graph f = A graph f A graph f A graph f, which implies for x [0, ], x, f x graph f x +, graph f x +, f x + graph f which i tur imply f x if x [ ] 0, f x = if x [, ] f x + if x [, ] dµ f x if x [ ] 0, dµ f x = 0 if x [, ] dµ f x if x [, ] This ca be used to fid the momets: m = x dµ f x = [0, x dµ f x + [, x dµ f x = [0, x dµ f x + [, x dµ f x x x + dµ f x so for, = dµf x + = m + m + m = = = m m 5
6 Usig this ad m 0 = 0, the first few are give by m =, m =, m 8 = 5, m 6 4 = Secod Example of Sigular Cotiuous Measure: Ferec Riesz The previous example is ot really that differet from the atomic case except the support of the sigular part of the measure is a ucoutable fractal set rather tha a coutable collectio of poits. To see how strage sigular measures ca be, cosider the followig costructio due to Ferec Riesz: Pick a t 0,. Start with the graph of F 0 beig the lie segmet from 0, 0 to,. To costruct the graph of F +, replace each lie segmet x 0, y 0 x, y i the graph of F with the pair of lie segmets x0 + x x 0, y 0, t y t y ad x0 + x, t y t y x, y I the limit as, the sequece of icreasig, cotiuous fuctios F =0 from the uit iterval oto itself coverges i sup orm to a odecreasig actually icreasig, cotiuous fuctio F : [0, ] [0, ]. This ca also be iterpreted as the affie iterated fractal system give by base B = 0, 0,, 0, 0, ad geerators G = 0, 0,, 0, 0, +t ad G =, +t,, +t,, with correspodig affie trasformatios A x, y = x, + t y ad A x, y = x +, t y + + t By repeated applicatio of the chai rule, for x =.b b b... i biary { df + t if b = 0 dx = x t if b = =0 At biary fractios, the two differet biary expasios correspod to takig the respective oe-sided derivatives. If lim sup umber of 0 s i first biary digits of x 6 < log t log + t log t
7 the df dx x exists ad is equal to 0. Yet, by applyig the cetral limit theorem to the biomial distributio it ca be show see below for details that, with respect to Lebesgue measure, for almost every x [0, ], so df dx umber of 0 s i first biary digits of x lim = = 0 almost everywhere, so the Rado-Nikodym derivative dµ F dλ = 0. Sice F is cotiuous, µ F has o atoms, so it is purely sigular cotiuous. Ulike i the previous example, however, the support of µ F is all of [0, ] sice those x with lim sup umber of 0 s i first biary digits of x log t log + t log t are a dese subset; hece, F is icreasig. The formulatio as a iterative fractal system ca also be used to do certai itegrals with respect to µ F. Sice graph F = A graph F A graph F which implies for x [0, ], x, + t F x graph F ad x +, t F x + + t graph F which i tur imply we have F x = dµ F x = { +t F x if x [ 0, +t F x + if x [, ] t { +t dµ F x if x [ 0, t dµ F x if x [, ] Usig this, we ca calcute the momets by gettig a relatio, m = 0 x dµ F x dx = 0 x + t dµ F x + 7 ] ] x t dµ F x
8 = + t The for 0 x dµf x + t m = + t m + + t m + + m = t + = Usig m 0 =, the first few are give by: m = t, m = t + t m 4 = t 0 x + dµ F x = m, m = t m + + t t 5t ad for t = : m = 4, m = 8, m = 7 4, m 4 = t t 7 4 Proof of Assertio I order to show that, with respect to Lebesgue measure, for almost every x [0, ], umber of 0 s i first biary digits of x lim it is oly ecessary to show that for ay p, ], the set of x [0, ] such that lim sup umber of 0 s i first biary digits of x has Lebesgue measure 0 sice by symmetry betwee 0 s ad s i the biary expasio, the correspodig set of x [0, ] such that lim if umber of 0 s i first biary digits of x = p p will also have Lebesgue measure 0, ad the, sice coutable uios of measure 0 sets are of measure 0, we ca simply apply the result to the sequece + for the values of p. = 8
9 Now take ay δ > 0 ad let q, p. The the proportio of biary fractios with deomiator that have proportio of 0 s i first biary digits greater tha or equal to q is give by = q For large, by applyig the cetral limit theorem to the biomial distributio, for of order from, π 4 e 4 = π e sice for = the mea is ad the variace. The same result also 4 follows usig Stirlig s formula. Ufortuately, we are iterested i of order from ; however, by umerical computatio the Gaussia is actually a rather drastic overestimate i this regio, so = q lettig x = = < q = q π e t < q π e dt e x dx < e x dx π q π t. Therefore, for sufficietly large, = q < erfc q e q π q from the asymptotic form for the complimetary error fuctio. Hece, there is a sequece of umbers, a =0 with the followig properties: i the sequece has limit ; ii the sum a =0 = q 9
10 is fiite. For example, from the precedig asymptotic expressio, a = r for r, e q will certaily work by the limit compariso test. Let the sum be deoted by M. Now cosider the subset B give by the uio of balls of diameter aδ M cetered at biary fractios with deomiator that have proportio of 0 s i their first biary digits greater tha or equal to q. The Lebesgue measure of B is less tha or equal to =0 a δ M = q = δ Sice a, there is a N such that for all m > N, am δ M umber such that lim sup umber of 0 s i first biary digits of x so by defiitio there is a > N such that >. Let x be ay p umber of 0 s i first biary digits of x q The x is withi < aδ of a biary fractio with deomiator M that has proportio of 0 s i first biary digits greater tha or equal to q, so x is i B. Sice x was arbitrary, all such umbers are i B. Sice δ was arbitrary, we ca coclude that the set of all such umbers is of Lebesgue measure 0. 0
Measure and Measurable Functions
3 Measure ad Measurable Fuctios 3.1 Measure o a Arbitrary σ-algebra Recall from Chapter 2 that the set M of all Lebesgue measurable sets has the followig properties: R M, E M implies E c M, E M for N implies
More informationProduct measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014.
Product measures, Toelli s ad Fubii s theorems For use i MAT3400/4400, autum 2014 Nadia S. Larse Versio of 13 October 2014. 1. Costructio of the product measure The purpose of these otes is to preset the
More informationLecture 3 The Lebesgue Integral
Lecture 3: The Lebesgue Itegral 1 of 14 Course: Theory of Probability I Term: Fall 2013 Istructor: Gorda Zitkovic Lecture 3 The Lebesgue Itegral The costructio of the itegral Uless expressly specified
More informationAxioms of Measure Theory
MATH 532 Axioms of Measure Theory Dr. Neal, WKU I. The Space Throughout the course, we shall let X deote a geeric o-empty set. I geeral, we shall ot assume that ay algebraic structure exists o X so that
More informationIntegrable Functions. { f n } is called a determining sequence for f. If f is integrable with respect to, then f d does exist as a finite real number
MATH 532 Itegrable Fuctios Dr. Neal, WKU We ow shall defie what it meas for a measurable fuctio to be itegrable, show that all itegral properties of simple fuctios still hold, ad the give some coditios
More informationThe Boolean Ring of Intervals
MATH 532 Lebesgue Measure Dr. Neal, WKU We ow shall apply the results obtaied about outer measure to the legth measure o the real lie. Throughout, our space X will be the set of real umbers R. Whe ecessary,
More informationMath 61CM - Solutions to homework 3
Math 6CM - Solutios to homework 3 Cédric De Groote October 2 th, 208 Problem : Let F be a field, m 0 a fixed oegative iteger ad let V = {a 0 + a x + + a m x m a 0,, a m F} be the vector space cosistig
More informationLecture Notes for Analysis Class
Lecture Notes for Aalysis Class Topological Spaces A topology for a set X is a collectio T of subsets of X such that: (a) X ad the empty set are i T (b) Uios of elemets of T are i T (c) Fiite itersectios
More informationIf a subset E of R contains no open interval, is it of zero measure? For instance, is the set of irrationals in [0, 1] is of measure zero?
2 Lebesgue Measure I Chapter 1 we defied the cocept of a set of measure zero, ad we have observed that every coutable set is of measure zero. Here are some atural questios: If a subset E of R cotais a
More informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
More informationSolution. 1 Solutions of Homework 1. Sangchul Lee. October 27, Problem 1.1
Solutio Sagchul Lee October 7, 017 1 Solutios of Homework 1 Problem 1.1 Let Ω,F,P) be a probability space. Show that if {A : N} F such that A := lim A exists, the PA) = lim PA ). Proof. Usig the cotiuity
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationChapter 0. Review of set theory. 0.1 Sets
Chapter 0 Review of set theory Set theory plays a cetral role i the theory of probability. Thus, we will ope this course with a quick review of those otios of set theory which will be used repeatedly.
More informationChapter 3. Strong convergence. 3.1 Definition of almost sure convergence
Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i
More informationAnalytic Continuation
Aalytic Cotiuatio The stadard example of this is give by Example Let h (z) = 1 + z + z 2 + z 3 +... kow to coverge oly for z < 1. I fact h (z) = 1/ (1 z) for such z. Yet H (z) = 1/ (1 z) is defied for
More informationLecture 3 : Random variables and their distributions
Lecture 3 : Radom variables ad their distributios 3.1 Radom variables Let (Ω, F) ad (S, S) be two measurable spaces. A map X : Ω S is measurable or a radom variable (deoted r.v.) if X 1 (A) {ω : X(ω) A}
More informationPRELIM PROBLEM SOLUTIONS
PRELIM PROBLEM SOLUTIONS THE GRAD STUDENTS + KEN Cotets. Complex Aalysis Practice Problems 2. 2. Real Aalysis Practice Problems 2. 4 3. Algebra Practice Problems 2. 8. Complex Aalysis Practice Problems
More informationArchimedes - numbers for counting, otherwise lengths, areas, etc. Kepler - geometry for planetary motion
Topics i Aalysis 3460:589 Summer 007 Itroductio Ree descartes - aalysis (breaig dow) ad sythesis Sciece as models of ature : explaatory, parsimoious, predictive Most predictios require umerical values,
More informationFUNDAMENTALS OF REAL ANALYSIS by
FUNDAMENTALS OF REAL ANALYSIS by Doğa Çömez Backgroud: All of Math 450/1 material. Namely: basic set theory, relatios ad PMI, structure of N, Z, Q ad R, basic properties of (cotiuous ad differetiable)
More informationMath Solutions to homework 6
Math 175 - Solutios to homework 6 Cédric De Groote November 16, 2017 Problem 1 (8.11 i the book): Let K be a compact Hermitia operator o a Hilbert space H ad let the kerel of K be {0}. Show that there
More informationMathematical Methods for Physics and Engineering
Mathematical Methods for Physics ad Egieerig Lecture otes Sergei V. Shabaov Departmet of Mathematics, Uiversity of Florida, Gaiesville, FL 326 USA CHAPTER The theory of covergece. Numerical sequeces..
More informationLecture 19: Convergence
Lecture 19: Covergece Asymptotic approach I statistical aalysis or iferece, a key to the success of fidig a good procedure is beig able to fid some momets ad/or distributios of various statistics. I may
More informationf(1), and so, if f is continuous, f(x) = f(1)x.
2.2.35: Let f be a additive fuctio. i Clearly fx = fx ad therefore f x = fx for all Z+ ad x R. Hece, for ay, Z +, f = f, ad so, if f is cotiuous, fx = fx. ii Suppose that f is bouded o soe o-epty ope set.
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More informationReal and Complex Analysis, 3rd Edition, W.Rudin
Real ad Complex Aalysis, 3rd ditio, W.Rudi Chapter 6 Complex Measures Yug-Hsiag Huag 206/08/22. Let ν be a complex measure o (X, M ). If M, defie { } µ () = sup ν( j ) : N,, 2, disjoit, = j { } ν () =
More informationThe Borel hierarchy classifies subsets of the reals by their topological complexity. Another approach is to classify them by size.
Lecture 7: Measure ad Category The Borel hierarchy classifies subsets of the reals by their topological complexity. Aother approach is to classify them by size. Filters ad Ideals The most commo measure
More informationProperties of Fuzzy Length on Fuzzy Set
Ope Access Library Joural 206, Volume 3, e3068 ISSN Olie: 2333-972 ISSN Prit: 2333-9705 Properties of Fuzzy Legth o Fuzzy Set Jehad R Kider, Jaafar Imra Mousa Departmet of Mathematics ad Computer Applicatios,
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More information5 Many points of continuity
Tel Aviv Uiversity, 2013 Measure ad category 40 5 May poits of cotiuity 5a Discotiuous derivatives.............. 40 5b Baire class 1 (classical)............... 42 5c Baire class 1 (moder)...............
More informationConvergence of random variables. (telegram style notes) P.J.C. Spreij
Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space
More informationChapter 6 Infinite Series
Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat
More informationREAL ANALYSIS II: PROBLEM SET 1 - SOLUTIONS
REAL ANALYSIS II: PROBLEM SET 1 - SOLUTIONS 18th Feb, 016 Defiitio (Lipschitz fuctio). A fuctio f : R R is said to be Lipschitz if there exists a positive real umber c such that for ay x, y i the domai
More informationHOMEWORK #4 - MA 504
HOMEWORK #4 - MA 504 PAULINHO TCHATCHATCHA Chapter 2, problem 19. (a) If A ad B are disjoit closed sets i some metric space X, prove that they are separated. (b) Prove the same for disjoit ope set. (c)
More informationMATH 413 FINAL EXAM. f(x) f(y) M x y. x + 1 n
MATH 43 FINAL EXAM Math 43 fial exam, 3 May 28. The exam starts at 9: am ad you have 5 miutes. No textbooks or calculators may be used durig the exam. This exam is prited o both sides of the paper. Good
More informationA sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as
More informationMcGill University Math 354: Honors Analysis 3 Fall 2012 Solutions to selected problems
McGill Uiversity Math 354: Hoors Aalysis 3 Fall 212 Assigmet 3 Solutios to selected problems Problem 1. Lipschitz fuctios. Let Lip K be the set of all fuctios cotiuous fuctios o [, 1] satisfyig a Lipschitz
More informationHere are some examples of algebras: F α = A(G). Also, if A, B A(G) then A, B F α. F α = A(G). In other words, A(G)
MATH 529 Probability Axioms Here we shall use the geeral axioms of a probability measure to derive several importat results ivolvig probabilities of uios ad itersectios. Some more advaced results will
More informationIt is always the case that unions, intersections, complements, and set differences are preserved by the inverse image of a function.
MATH 532 Measurable Fuctios Dr. Neal, WKU Throughout, let ( X, F, µ) be a measure space ad let (!, F, P ) deote the special case of a probability space. We shall ow begi to study real-valued fuctios defied
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 19 11/17/2008 LAWS OF LARGE NUMBERS II THE STRONG LAW OF LARGE NUMBERS
MASSACHUSTTS INSTITUT OF TCHNOLOGY 6.436J/5.085J Fall 2008 Lecture 9 /7/2008 LAWS OF LARG NUMBRS II Cotets. The strog law of large umbers 2. The Cheroff boud TH STRONG LAW OF LARG NUMBRS While the weak
More informationSequences A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece 1, 1, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet
More informationMATH301 Real Analysis (2008 Fall) Tutorial Note #7. k=1 f k (x) converges pointwise to S(x) on E if and
MATH01 Real Aalysis (2008 Fall) Tutorial Note #7 Sequece ad Series of fuctio 1: Poitwise Covergece ad Uiform Covergece Part I: Poitwise Covergece Defiitio of poitwise covergece: A sequece of fuctios f
More information7.1 Convergence of sequences of random variables
Chapter 7 Limit Theorems Throughout this sectio we will assume a probability space (, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite
More informationA Proof of Birkhoff s Ergodic Theorem
A Proof of Birkhoff s Ergodic Theorem Joseph Hora September 2, 205 Itroductio I Fall 203, I was learig the basics of ergodic theory, ad I came across this theorem. Oe of my supervisors, Athoy Quas, showed
More information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More information6. Uniform distribution mod 1
6. Uiform distributio mod 1 6.1 Uiform distributio ad Weyl s criterio Let x be a seuece of real umbers. We may decompose x as the sum of its iteger part [x ] = sup{m Z m x } (i.e. the largest iteger which
More informationABOUT CHAOS AND SENSITIVITY IN TOPOLOGICAL DYNAMICS
ABOUT CHAOS AND SENSITIVITY IN TOPOLOGICAL DYNAMICS EDUARD KONTOROVICH Abstract. I this work we uify ad geeralize some results about chaos ad sesitivity. Date: March 1, 005. 1 1. Symbolic Dyamics Defiitio
More informationAdvanced Stochastic Processes.
Advaced Stochastic Processes. David Gamarik LECTURE 2 Radom variables ad measurable fuctios. Strog Law of Large Numbers (SLLN). Scary stuff cotiued... Outlie of Lecture Radom variables ad measurable fuctios.
More informationMath 220A Fall 2007 Homework #2. Will Garner A
Math 0A Fall 007 Homewor # Will Garer Pg 3 #: Show that {cis : a o-egative iteger} is dese i T = {z œ : z = }. For which values of q is {cis(q): a o-egative iteger} dese i T? To show that {cis : a o-egative
More informationThe Pointwise Ergodic Theorem and its Applications
The Poitwise Ergodic Theorem ad its Applicatios Itroductio Peter Oberly 11/9/2018 Algebra has homomorphisms ad topology has cotiuous maps; i these otes we explore the structure preservig maps for measure
More informationMath 341 Lecture #31 6.5: Power Series
Math 341 Lecture #31 6.5: Power Series We ow tur our attetio to a particular kid of series of fuctios, amely, power series, f(x = a x = a 0 + a 1 x + a 2 x 2 + where a R for all N. I terms of a series
More informationFUNDAMENTALS OF REAL ANALYSIS by. V.1. Product measures
FUNDAMENTALS OF REAL ANALSIS by Doğa Çömez V. PRODUCT MEASURE SPACES V.1. Product measures Let (, A, µ) ad (, B, ν) be two measure spaces. I this sectio we will costruct a product measure µ ν o that coicides
More informationTheorem 3. A subset S of a topological space X is compact if and only if every open cover of S by open sets in X has a finite subcover.
Compactess Defiitio 1. A cover or a coverig of a topological space X is a family C of subsets of X whose uio is X. A subcover of a cover C is a subfamily of C which is a cover of X. A ope cover of X is
More informationMath 140A Elementary Analysis Homework Questions 3-1
Math 0A Elemetary Aalysis Homework Questios -.9 Limits Theorems for Sequeces Suppose that lim x =, lim y = 7 ad that all y are o-zero. Detarime the followig limits: (a) lim(x + y ) (b) lim y x y Let s
More informationPart A, for both Section 200 and Section 501
Istructios Please write your solutios o your ow paper. These problems should be treated as essay questios. A problem that says give a example or determie requires a supportig explaatio. I all problems,
More information1 Introduction. 1.1 Notation and Terminology
1 Itroductio You have already leared some cocepts of calculus such as limit of a sequece, limit, cotiuity, derivative, ad itegral of a fuctio etc. Real Aalysis studies them more rigorously usig a laguage
More informationChapter 3 Inner Product Spaces. Hilbert Spaces
Chapter 3 Ier Product Spaces. Hilbert Spaces 3. Ier Product Spaces. Hilbert Spaces 3.- Defiitio. A ier product space is a vector space X with a ier product defied o X. A Hilbert space is a complete ier
More information7 Sequences of real numbers
40 7 Sequeces of real umbers 7. Defiitios ad examples Defiitio 7... A sequece of real umbers is a real fuctio whose domai is the set N of atural umbers. Let s : N R be a sequece. The the values of s are
More informationAbstract Vector Spaces. Abstract Vector Spaces
Astract Vector Spaces The process of astractio is critical i egieerig! Physical Device Data Storage Vector Space MRI machie Optical receiver 0 0 1 0 1 0 0 1 Icreasig astractio 6.1 Astract Vector Spaces
More informationn=1 a n is the sequence (s n ) n 1 n=1 a n converges to s. We write a n = s, n=1 n=1 a n
Series. Defiitios ad first properties A series is a ifiite sum a + a + a +..., deoted i short by a. The sequece of partial sums of the series a is the sequece s ) defied by s = a k = a +... + a,. k= Defiitio
More informationProbability and Statistics
robability ad Statistics rof. Zheg Zheg Radom Variable A fiite sigle valued fuctio.) that maps the set of all eperimetal outcomes ito the set of real umbers R is a r.v., if the set ) is a evet F ) for
More information1 Convergence in Probability and the Weak Law of Large Numbers
36-752 Advaced Probability Overview Sprig 2018 8. Covergece Cocepts: i Probability, i L p ad Almost Surely Istructor: Alessadro Rialdo Associated readig: Sec 2.4, 2.5, ad 4.11 of Ash ad Doléas-Dade; Sec
More informationDefinition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4.
4. BASES I BAACH SPACES 39 4. BASES I BAACH SPACES Sice a Baach space X is a vector space, it must possess a Hamel, or vector space, basis, i.e., a subset {x γ } γ Γ whose fiite liear spa is all of X ad
More informationPAPER : IIT-JAM 2010
MATHEMATICS-MA (CODE A) Q.-Q.5: Oly oe optio is correct for each questio. Each questio carries (+6) marks for correct aswer ad ( ) marks for icorrect aswer.. Which of the followig coditios does NOT esure
More information7.1 Convergence of sequences of random variables
Chapter 7 Limit theorems Throughout this sectio we will assume a probability space (Ω, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite
More informationChapter 7 Isoperimetric problem
Chapter 7 Isoperimetric problem Recall that the isoperimetric problem (see the itroductio its coectio with ido s proble) is oe of the most classical problem of a shape optimizatio. It ca be formulated
More informationSequences. A Sequence is a list of numbers written in order.
Sequeces A Sequece is a list of umbers writte i order. {a, a 2, a 3,... } The sequece may be ifiite. The th term of the sequece is the th umber o the list. O the list above a = st term, a 2 = 2 d term,
More informationAn Introduction to Randomized Algorithms
A Itroductio to Radomized Algorithms The focus of this lecture is to study a radomized algorithm for quick sort, aalyze it usig probabilistic recurrece relatios, ad also provide more geeral tools for aalysis
More information3 Gauss map and continued fractions
ICTP, Trieste, July 08 Gauss map ad cotiued fractios I this lecture we will itroduce the Gauss map, which is very importat for its coectio with cotiued fractios i umber theory. The Gauss map G : [0, ]
More informationB Supplemental Notes 2 Hypergeometric, Binomial, Poisson and Multinomial Random Variables and Borel Sets
B671-672 Supplemetal otes 2 Hypergeometric, Biomial, Poisso ad Multiomial Radom Variables ad Borel Sets 1 Biomial Approximatio to the Hypergeometric Recall that the Hypergeometric istributio is fx = x
More informationIn this section, we show how to use the integral test to decide whether a series
Itegral Test Itegral Test Example Itegral Test Example p-series Compariso Test Example Example 2 Example 3 Example 4 Example 5 Exa Itegral Test I this sectio, we show how to use the itegral test to decide
More informationLesson 10: Limits and Continuity
www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals
More informationSequences and Series of Functions
Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges
More informationDANIELL AND RIEMANN INTEGRABILITY
DANIELL AND RIEMANN INTEGRABILITY ILEANA BUCUR We itroduce the otio of Riema itegrable fuctio with respect to a Daiell itegral ad prove the approximatio theorem of such fuctios by a mootoe sequece of Jorda
More information1.3 Convergence Theorems of Fourier Series. k k k k. N N k 1. With this in mind, we state (without proof) the convergence of Fourier series.
.3 Covergece Theorems of Fourier Series I this sectio, we preset the covergece of Fourier series. A ifiite sum is, by defiitio, a limit of partial sums, that is, a cos( kx) b si( kx) lim a cos( kx) b si(
More informationANSWERS TO MIDTERM EXAM # 2
MATH 03, FALL 003 ANSWERS TO MIDTERM EXAM # PENN STATE UNIVERSITY Problem 1 (18 pts). State ad prove the Itermediate Value Theorem. Solutio See class otes or Theorem 5.6.1 from our textbook. Problem (18
More informationSequence A sequence is a function whose domain of definition is the set of natural numbers.
Chapter Sequeces Course Title: Real Aalysis Course Code: MTH3 Course istructor: Dr Atiq ur Rehma Class: MSc-I Course URL: wwwmathcityorg/atiq/fa8-mth3 Sequeces form a importat compoet of Mathematical Aalysis
More informationHOMEWORK #10 SOLUTIONS
Math 33 - Aalysis I Sprig 29 HOMEWORK # SOLUTIONS () Prove that the fuctio f(x) = x 3 is (Riema) itegrable o [, ] ad show that x 3 dx = 4. (Without usig formulae for itegratio that you leart i previous
More informationIntroductory Analysis I Fall 2014 Homework #7 Solutions
Itroductory Aalysis I Fall 214 Homework #7 Solutios Note: There were a couple of typos/omissios i the formulatio of this homework. Some of them were, I believe, quite obvious. The fact that the statemet
More informationMATH 31B: MIDTERM 2 REVIEW
MATH 3B: MIDTERM REVIEW JOE HUGHES. Evaluate x (x ) (x 3).. Partial Fractios Solutio: The umerator has degree less tha the deomiator, so we ca use partial fractios. Write x (x ) (x 3) = A x + A (x ) +
More information17. Joint distributions of extreme order statistics Lehmann 5.1; Ferguson 15
17. Joit distributios of extreme order statistics Lehma 5.1; Ferguso 15 I Example 10., we derived the asymptotic distributio of the maximum from a radom sample from a uiform distributio. We did this usig
More informationMath 21B-B - Homework Set 2
Math B-B - Homework Set Sectio 5.:. a) lim P k= c k c k ) x k, where P is a partitio of [, 5. x x ) dx b) lim P k= 4 ck x k, where P is a partitio of [,. 4 x dx c) lim P k= ta c k ) x k, where P is a partitio
More informations = and t = with C ij = A i B j F. (i) Note that cs = M and so ca i µ(a i ) I E (cs) = = c a i µ(a i ) = ci E (s). (ii) Note that s + t = M and so
3 From the otes we see that the parts of Theorem 4. that cocer us are: Let s ad t be two simple o-egative F-measurable fuctios o X, F, µ ad E, F F. The i I E cs ci E s for all c R, ii I E s + t I E s +
More informationAdvanced Analysis. Min Yan Department of Mathematics Hong Kong University of Science and Technology
Advaced Aalysis Mi Ya Departmet of Mathematics Hog Kog Uiversity of Sciece ad Techology September 3, 009 Cotets Limit ad Cotiuity 7 Limit of Sequece 8 Defiitio 8 Property 3 3 Ifiity ad Ifiitesimal 8 4
More informationWe are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n
Review of Power Series, Power Series Solutios A power series i x - a is a ifiite series of the form c (x a) =c +c (x a)+(x a) +... We also call this a power series cetered at a. Ex. (x+) is cetered at
More information2 Banach spaces and Hilbert spaces
2 Baach spaces ad Hilbert spaces Tryig to do aalysis i the ratioal umbers is difficult for example cosider the set {x Q : x 2 2}. This set is o-empty ad bouded above but does ot have a least upper boud
More informationFall 2013 MTH431/531 Real analysis Section Notes
Fall 013 MTH431/531 Real aalysis Sectio 8.1-8. Notes Yi Su 013.11.1 1. Defiitio of uiform covergece. We look at a sequece of fuctios f (x) ad study the coverget property. Notice we have two parameters
More informationMA541 : Real Analysis. Tutorial and Practice Problems - 1 Hints and Solutions
MA54 : Real Aalysis Tutorial ad Practice Problems - Hits ad Solutios. Suppose that S is a oempty subset of real umbers that is bouded (i.e. bouded above as well as below). Prove that if S sup S. What ca
More informationNotes 27 : Brownian motion: path properties
Notes 27 : Browia motio: path properties Math 733-734: Theory of Probability Lecturer: Sebastie Roch Refereces:[Dur10, Sectio 8.1], [MP10, Sectio 1.1, 1.2, 1.3]. Recall: DEF 27.1 (Covariace) Let X = (X
More informationCouncil for Innovative Research
ABSTRACT ON ABEL CONVERGENT SERIES OF FUNCTIONS ERDAL GÜL AND MEHMET ALBAYRAK Yildiz Techical Uiversity, Departmet of Mathematics, 34210 Eseler, Istabul egul34@gmail.com mehmetalbayrak12@gmail.com I this
More informationLecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
More informationM17 MAT25-21 HOMEWORK 5 SOLUTIONS
M17 MAT5-1 HOMEWORK 5 SOLUTIONS 1. To Had I Cauchy Codesatio Test. Exercise 1: Applicatio of the Cauchy Codesatio Test Use the Cauchy Codesatio Test to prove that 1 diverges. Solutio 1. Give the series
More informationLecture 6: Integration and the Mean Value Theorem. slope =
Math 8 Istructor: Padraic Bartlett Lecture 6: Itegratio ad the Mea Value Theorem Week 6 Caltech 202 The Mea Value Theorem The Mea Value Theorem abbreviated MVT is the followig result: Theorem. Suppose
More informationNotes #3 Sequences Limit Theorems Monotone and Subsequences Bolzano-WeierstraßTheorem Limsup & Liminf of Sequences Cauchy Sequences and Completeness
Notes #3 Sequeces Limit Theorems Mootoe ad Subsequeces Bolzao-WeierstraßTheorem Limsup & Limif of Sequeces Cauchy Sequeces ad Completeess This sectio of otes focuses o some of the basics of sequeces of
More information4x 2. (n+1) x 3 n+1. = lim. 4x 2 n+1 n3 n. n 4x 2 = lim = 3
Exam Problems (x. Give the series (, fid the values of x for which this power series coverges. Also =0 state clearly what the radius of covergece is. We start by settig up the Ratio Test: x ( x x ( x x
More informationChapter 2 The Monte Carlo Method
Chapter 2 The Mote Carlo Method The Mote Carlo Method stads for a broad class of computatioal algorithms that rely o radom sampligs. It is ofte used i physical ad mathematical problems ad is most useful
More informationENGI Series Page 6-01
ENGI 3425 6 Series Page 6-01 6. Series Cotets: 6.01 Sequeces; geeral term, limits, covergece 6.02 Series; summatio otatio, covergece, divergece test 6.03 Stadard Series; telescopig series, geometric series,
More informationINTEGRATION THEORY AND FUNCTIONAL ANALYSIS MM-501
INTEGRATION THEORY AND FUNCTIONAL ANALYSIS M.A./M.Sc. Mathematics (Fial) MM-50 Directorate of Distace Educatio Maharshi Dayaad Uiversity ROHTAK 4 00 Copyright 004, Maharshi Dayaad Uiversity, ROHTAK All
More informationRandom Variables, Sampling and Estimation
Chapter 1 Radom Variables, Samplig ad Estimatio 1.1 Itroductio This chapter will cover the most importat basic statistical theory you eed i order to uderstad the ecoometric material that will be comig
More informationFinal Solutions. 1. (25pts) Define the following terms. Be as precise as you can.
Mathematics H104 A. Ogus Fall, 004 Fial Solutios 1. (5ts) Defie the followig terms. Be as recise as you ca. (a) (3ts) A ucoutable set. A ucoutable set is a set which ca ot be ut ito bijectio with a fiite
More informationChapter 10: Power Series
Chapter : Power Series 57 Chapter Overview: Power Series The reaso series are part of a Calculus course is that there are fuctios which caot be itegrated. All power series, though, ca be itegrated because
More information