Entrance Exam, Real Analysis September 1, 2017 Solve exactly 6 out of the 8 problems
|
|
- Molly Stevens
- 6 years ago
- Views:
Transcription
1 September, 27 Solve exactly 6 out of the 8 problems. Prove by denition (in ɛ δ language) that f(x) = + x 2 is uniformly continuous in (, ). Is f(x) uniformly continuous in (, )? Prove your conclusion. 2. Let f n (x) = nx n + x. (a) Find f(x) = f n (x); (b) Does f n converge to f uniformly (, )? Prove your conclusion. (c) Does f n converge to f uniformly (, )? Prove your conclusion. 3. If s n, which of the following statements are true? Explain your conclusion. (a) N > such that n > N, s n. (b) N >, n > N such that s n. (c) ɛ > such that N >, n > N with s n. (d) s2 n ( s n) Let g(x) be dened as follows g(x) = (a) Prove g(x) is continuous in (, ). cos(xy 3 ) + y 2 dy. (b) Is g(x) uniformly continuous in (, )? absolutely continuous in (, )? Prove your conclusion. 5. Let {a n } n= be a sequence of real numbers satisfying n= a n <. Dene f by f(x) = n= a nx n. (a) Show that f is a function of bounded variation on x [, ]. (b) If a and a n = for n 2, is f absolutely continuous on x R. Explain your answer. (c) If a, a 2, and a n = for n 3, is f absolutely continuous on x R. Explain your answer.
2 6. For p < dene f p = ( f p ) /p E and denote L p (E) to be the set of functions f for which E f p <. Either prove the statement or show a counter example. (a) If E = [, ], then there is a constant c > for which f c f 2 for all f L 2 (E). (b) Suppose E = [, ). L (E). If f is bounded and E f 2 <, then it belongs to 7. Suppose that f(x) is a uniformly continuous and Lebesgue integrable function on R. Show that f(x) =. x 8. Let {u n } n= be a sequence of Lebesgue measurable functions on [, ] and assume u n (x) = a.e. on [, ], and also u n L 2 [,] for all n. Prove that u n L [,] =. 2
3 April 2, 27 Solve exactly 6 out of the 8 problems. Is f(x) = x 3 uniformly continuous in (-5, )? in (, )? Prove your conclusion. 2. Let f n (x) = nx n + x. (a) Find f(x) = f n (x); (b) Does f n converge to f uniformly (, )? Prove your conclusion. (c) Does f n converge to f uniformly (, )? Prove your conclusion. 3. If s n, which of the following statements are true? Explain your conclusion. (a) N > such that n > N, s n. (b) N >, n > N such that s n. (c) ɛ > such that N >, n > N with s n > ɛ. (d) s2 n > ( s n) Let g(x) be dened as follows g(x) = (a) Prove g(x) is continuous in (, ). cos 2 (xy) + y 2 dy. (b) Is g(x) uniformly continuous in (, )? absolutely continuous in (, )? Prove your conclusion. 5. Let {a n } n= be a sequence of real numbers satisfying n= a n <. Dene f : [, ] R by f(x) = n= a nx n. Show that f is a function of bounded variation. 6. Suppose that f : [, ) [, ) is measurable and that f(x) <. Prove that x n f(x) + x n dx = f(x)dx 7. Suppose that f(x) is a uniformly continuous and Lebesgue integrable function on R. Show that f(x) =. x 8. Let {u n } n= be a sequence of Lebesgue measurable functions on [, ] and assume u n (x) = a.e. on [, ], and also u n L 2 [,] for all n. Prove that u n L [,] =.
4 August 3, 26 Solve exactly 6 out of the 8 problems. (a) Prove by denition that x 2 is uniformly continuous in ( 7, 2). (b) Prove by denition that x 2 is not uniformly continuous in [, ). 2. If s n =, which of the following statements are true? Explain your conclusion. (a) N > such that n > N, s n. (b) N >, n > N such that s n. (c) ɛ >, N >, such that n > N, s n > ɛ. (d) ɛ > such that N >, n > N with s n > ɛ. 3. Let f(x) be a function of bounded variation dened on [,]. Prove that the set of all discontinuity points of f(x) is countable. 4. Let f(x) be a measurable function dened in R and satisfying: (a) f() = ; (b) If f(x ) >, then δ > such thatf(x) > in (x δ, x + δ); (c) If f(x n ) > and x n a, then f(a) > ). Is it true f(x) > x R? Prove your conclusion. 5. Suppose that {f n } is a sequence of real valued measurable functions dened on the interval [, ] and suppose that f n(x) = f(x) a.e. on [, ]. Let p >, M >, and that f n L p (R) M for all n. (a) Prove that f L p (R) and that f L p (R) M. (b) Prove that f f n L p (R) =. 6. Suppose that f L (R) is a uniformly continuous function. Show that f(x) =. x
5 7. For p < dene f p = ( f p ) /p E and denote L p (E) to be the set of functions f for which E f p <. statement below, either prove it or show a counter example. For each (a) If E = [, ], then there is a constant c for which f 2 c f for all f L (E). (b) Suppose E = [, ). If f is bounded and E f 2 <, then it belongs to L (E). 8. (a) For a Lebesgue integrable function f on [,] dene F (x) = x f(t)dt for x [, ]. Prove that F is absolutely continuous on [,]. (b) Give an explicit example of a continuous function F of bounded variation on [,] that is not absolutely continuous on [,]. It is not necessary to verify your assertion. 2
6 April 2, 26 Solve exactly 6 out of the 8 problems. Let f be a monotone function dened on [,] with f() = and f() =. Show that f has at most countable discontinuous points. 2. Answer the following questions and prove your conclusion. (a) Is x 2 uniformly continuous in the open interval ( 5, 2)? (b) Is x 2 uniformly continuous in [, )? 3. If s n >, which of the following statements are true? Prove your conclusion. (a) N > such that n > N, s n >. (b) ɛ > such that N >, n > N with s n > ɛ. (c) ɛ >, N >, such that n > N, s n > ɛ. (d) s2 n = ( s n) Let f L (, ), and g(x) be dened as follows g(x) = ˆ (a) Prove g(x) is continuous in (, ). cos 2 (xy)f(y)dy. (b) Is g(x) uniformly continuous in (, )? absolutely continuous in (, )? Prove your conclusion. 5. Let f and g be real valued measurable functions on [, ] with the property that g is dierentiable at every x on [, ] and Show that f L [, ]. g (x) = (f(x)) (a) Let m be the Lebesgue measure on R and let E R be measurable. Also, suppose f L (E) and that f > a.e. on E. Show that ˆ f(x) /n dx = m(e). E (b) Let E = [a, b], where a and b are real numbers satisfying a < b. With the same assumptions on f as in (a), show that and (ˆ b f(x) /n dx) n = if b a > a (ˆ b f(x) /n dx) n = if b a <. a
7 7. Suppose that f : [, ) [, ) is measurable and that f(x) <. Prove that ˆ x n ˆ f(x) + x n dx = f(x)dx 8. Let {u n } n= be a sequence of Lebesgue measurable functions on [, ] and assume u n (x) = a.e. on [, ], and also u n L 2 [,] for all n. Prove that u n L [,] =. 2
8 April 7, 24 Solve exactly 6 out of the 8 problems.. Answer the following questions and prove your conclusion. If A is a closed subset of [,] and A [, ]. Is it possible that ma =? If B is an open subset of [,] and dense in [, ]. Is it possible that mb <? 2. Let {p k } be all the rational numbers in [,], and H(x) be the Heaviside function defined as {, x >, H(x) =, x. Define f(x) in [,] as f(x) = ( ) k 2 k H(x p k ). k= Prove by definition (in ɛ δ language) that f(x) is continuous at all irrational numbers, and discontinuous at all rational numbers in [,]. Determine and explain: (i). Is f(x) Riemann integrable in [,]? (ii). Is f(x) Lebegue integrable? (iii) Is f(x) of bounded total variation? 3. Let f be Lebegue integrable in [,b] with b <. Prove b f(x) cos nx dx =. Is it also true if b =? Prove your conclusion. 4. If f is absolutely continuous on [,] and f(x) > on [,]. Prove by definition that /f is absolutely continuous on [,]. Is f(x) = x absolutely continuous on [,]? In [, )? Prove your conclusion. 5. Prove or disprove that for every ε > and for every f L [a, b], where a and b are finite, there is a g C[a, b] such that 6. Compute the Lebesgue integral ess sup a x b f g < ε. inf xex sin 2 (πnx) dx.
9 7. Prove or disprove the following statement. (a) Let g be an integrable function on [; ]. Then there is a bounded measurable function f such that fg = g f. (b) Let {f n } be a sequence of functions in L [, ], which converge almost everywhere to a function f in L, and suppose that there is a constant M such that f n M all n. Then for each function g in L we have fg = 8. Let {a n } n= be a sequence of real numbers satisfying n= a n <. Define f : [, ] R by f(x) = n= a2 nx n. Show that f is a function of bounded variation in [, ]. f n g. 2
10 April 22, 25 Solve exactly 6 out of the 8 problems.. Let A R be an open set with the following property: {x n } A, a subsequence {x nk } such that x nk x A. Which of the following statements is true? ma =? ma = ln 2? ma = π? ma =? ma is not uniquely determined and could be any positive number? Prove your conclusion. 2. Answer the following questions and prove your conclusion. Let A [, ] be an open set. If δ < such that ma δ, is it possible that A is dense in [,]? Let B [, ] and B [, ]. If ɛ >, mb > ɛ, is it possible that B is closed? 3. Let f L (, ), and g(x) be defined as follows g(x) = ˆ Prove g(x) is continuous in (, ). cos(xy)f(y)dy. Is g(x) uniformly continuous in (, )? Prove your conclusion. 4. Let f(x) > and be absolutely continuous on [,]. Let g(x) = is absolutely continuous on [,]. f(x). Prove g(x) 5. (a) For a bounded Lebesgue integrable function f on [,] define F (x) = x f(t)dt for x [, ]. Prove that F is absolutely continuous on [,]. (b) Give an explicit example of a continuous function F of bounded variation on [,] that is not absolutely continuous on [,]. It is not necessary to verify your assertion.
11 6. (a) Suppose that f L [, ] and let m denote Lebesgue measure. Prove that for c > m{ f(x) c} c ˆ f(x) dx. (b) Suppose that {f n } L p [, ] is a sequence of functions satisfying f n L p [,] M, where < p <. Prove that ˆ f n (x) dx =, uniformly in n. c f n(x) c 7. Assume that n is an integer and let f n= Ln [, ]. Prove that if then f = a.e. f L n [,] <, n= 8. Suppose that {f n } is a sequence in L (R) with f n L (R) for all n and f n(x) = f(x) a.e. (a) Prove that f L (R) and that f L (R). (b) Show that ( f f n L (R) f n L (R) + f L (R) ) =. Hint: The following inequality might be useful. For any numbers a and b a b a + b 2 b 2
12 August 27, 23 Solve exactly 6 out of the 8 problems. Given function φ(x) with φ() = and φ (x) = e x2. The plane curve Γ is dened by the equation y = φ( + xy + y 2 ). Find the equation for the tangent line to the curve Γ at the point (x, y) = (, ). 2. Does ( ) x f n (x) = x n sin x converge uniformly on (,)? Prove your conclusion. 3. Let {x n = p n /q n Q} be a sequence of rational numbers and x n α R\Q (α is irrational). Prove q n. 4. Let E be a Lebesgue measurable set in R, and f n be a sequence of nonnegative Lebesgue integrable functions on E. If f n f a.e. on E, is it always true that ˆ ˆ fdx f n dx? Prove your conclusion. E 5. Let a and b are real numbers satisfying a < b. Prove or disprove the following. (a) If f(x) is a dierentiable function on a < x < b, it is absolutely continuous on a < x < b. (b) If f(x) is absolutely continuous on a x b, it is Lipschitz on a x b. 6. A family F of measurable functions on E of nite measure is said to be uniformly integrable over E provided that for each ε >, there is a δ > such that for each f F and for each measurable set A E with ma < δ, A f < ε. Either prove or disprove the following statements. (a) If G is the family of measurable functions f on [, ], each of which is integrable over [, ] and has f <, then G is uniformly integrable over [, ]. (b) If {f n } is a sequence of nonnegative, measurable, and integrable functions that converges pointwise a.e. on E to h =, then {f n } is uniformly integrable. E
13 7. For p < dene ˆ f p = ( f p ) /p E and denote L p (E) to be the set of functions f for which E f p <. Either prove the statement or show a counter example. (a) If E = [, ], then there is a constant c for which f 2 c f for all f L (E). (b) Suppose E = [, ). If f is bounded and E f 2 <, then it belongs to L (E). 8. Prove or disprove that for every ε > and for every f L [a, b], where a and b are nite, there is a g C[a, b] such that ess sup a x b f g < ε. 2
14 April 22, 23 Solve exactly 6 out of the 8 problems. Given function φ(x) with φ() = 5 and φ (x) = e x2. The plane curve Γ is dened by the equation y = φ( + sin 3(xy)). Find the equation for the tangent line to the curve Γ at the point (x, y) = (, 5). 2. Does ( ) x f n (x) = x n sin x converge uniformly on (,)? Prove your conclusion. 3. Evaluate the integral I = ˆ dy ˆ y e x2 dx. 4. Let f, g be absolutely continuous in (,). Is it true that the product fg is also absolutely continuous in (,)? Prove your conclusion. 5. Let {f n } be a sequence of nonnegative integrable functions on E. For each of (a), (b), and (c) either prove the statement or show a counter example. (a) If E f n =, then {f n } in measure. (b) If {f n } in measure, then E f n =. (c) If {f n } almost everywhere on E, then E f n =. 6. For f in C(E), where C(E) is the set of continuous functions on E, dene ˆ f p = ( f p ) /p, f sup = sup f(x). x E E Here p <. For each of (a), (b), and (c) either prove the statement or show a counter example. (a) If E = [, ] and p < q <, then there is a constant c for which f p c f q for all f C(E). (b) If E = [, ], then there is a constant c for which f sup c f for all f C(E). (c) Suppose E = [, ). If f C(E) is bounded and E f 2 <, then it belongs to L (E).
15 7. Prove or disprove that for every ε > and for every f L [a, b], there is a g C[a, b] such that ess sup a x b f g < ε. 8. (a) Suppose E R has Lebesgue measure zero. Prove that if f : R R is increasing and absolutely continuous, then f(e) has Lebesgue measure zero. Here, f(e) = {y y = f(x) for some member x of E}. (b) Would the same conclusion hold if we remove increasing in the assumption on f? In other words, if f : R R is absolutely continuous, then would f(e) have Lebesgue measure zero? Explain your reasoning. 2
A List of Problems in Real Analysis
A List of Problems in Real Analysis W.Yessen & T.Ma December 3, 218 This document was first created by Will Yessen, who was a graduate student at UCI. Timmy Ma, who was also a graduate student at UCI,
More informationChapter 5. Measurable Functions
Chapter 5. Measurable Functions 1. Measurable Functions Let X be a nonempty set, and let S be a σ-algebra of subsets of X. Then (X, S) is a measurable space. A subset E of X is said to be measurable if
More informationAnalysis Qualifying Exam
Analysis Qualifying Exam Spring 2017 Problem 1: Let f be differentiable on R. Suppose that there exists M > 0 such that f(k) M for each integer k, and f (x) M for all x R. Show that f is bounded, i.e.,
More informationExistence of a Limit on a Dense Set, and. Construction of Continuous Functions on Special Sets
Existence of a Limit on a Dense Set, and Construction of Continuous Functions on Special Sets REU 2012 Recap: Definitions Definition Given a real-valued function f, the limit of f exists at a point c R
More information3 (Due ). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?
MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due ). Show that the open disk x 2 + y 2 < 1 is a countable union of planar elementary sets. Show that the closed disk x 2 + y 2 1 is a countable
More informationh(x) lim H(x) = lim Since h is nondecreasing then h(x) 0 for all x, and if h is discontinuous at a point x then H(x) > 0. Denote
Real Variables, Fall 4 Problem set 4 Solution suggestions Exercise. Let f be of bounded variation on [a, b]. Show that for each c (a, b), lim x c f(x) and lim x c f(x) exist. Prove that a monotone function
More informationMathematical Methods for Physics and Engineering
Mathematical Methods for Physics and Engineering Lecture notes for PDEs Sergei V. Shabanov Department of Mathematics, University of Florida, Gainesville, FL 32611 USA CHAPTER 1 The integration theory
More informationReal Analysis Problems
Real Analysis Problems Cristian E. Gutiérrez September 14, 29 1 1 CONTINUITY 1 Continuity Problem 1.1 Let r n be the sequence of rational numbers and Prove that f(x) = 1. f is continuous on the irrationals.
More informationSolutions Final Exam May. 14, 2014
Solutions Final Exam May. 14, 2014 1. (a) (10 points) State the formal definition of a Cauchy sequence of real numbers. A sequence, {a n } n N, of real numbers, is Cauchy if and only if for every ɛ > 0,
More information1. Let A R be a nonempty set that is bounded from above, and let a be the least upper bound of A. Show that there exists a sequence {a n } n N
Applied Analysis prelim July 15, 216, with solutions Solve 4 of the problems 1-5 and 2 of the problems 6-8. We will only grade the first 4 problems attempted from1-5 and the first 2 attempted from problems
More informationHomework 11. Solutions
Homework 11. Solutions Problem 2.3.2. Let f n : R R be 1/n times the characteristic function of the interval (0, n). Show that f n 0 uniformly and f n µ L = 1. Why isn t it a counterexample to the Lebesgue
More informationProbability and Measure
Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 84 Paper 4, Section II 26J Let (X, A) be a measurable space. Let T : X X be a measurable map, and µ a probability
More information2 (Bonus). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?
MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due 9/5). Prove that every countable set A is measurable and µ(a) = 0. 2 (Bonus). Let A consist of points (x, y) such that either x or y is
More informationSolutions: Problem Set 4 Math 201B, Winter 2007
Solutions: Problem Set 4 Math 2B, Winter 27 Problem. (a Define f : by { x /2 if < x
More information1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer 11(2) (1989),
Real Analysis 2, Math 651, Spring 2005 April 26, 2005 1 Real Analysis 2, Math 651, Spring 2005 Krzysztof Chris Ciesielski 1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer
More informationProblem Set 5: Solutions Math 201A: Fall 2016
Problem Set 5: s Math 21A: Fall 216 Problem 1. Define f : [1, ) [1, ) by f(x) = x + 1/x. Show that f(x) f(y) < x y for all x, y [1, ) with x y, but f has no fixed point. Why doesn t this example contradict
More informationExercise 1. Let f be a nonnegative measurable function. Show that. where ϕ is taken over all simple functions with ϕ f. k 1.
Real Variables, Fall 2014 Problem set 3 Solution suggestions xercise 1. Let f be a nonnegative measurable function. Show that f = sup ϕ, where ϕ is taken over all simple functions with ϕ f. For each n
More informationAustin Mohr Math 704 Homework 6
Austin Mohr Math 704 Homework 6 Problem 1 Integrability of f on R does not necessarily imply the convergence of f(x) to 0 as x. a. There exists a positive continuous function f on R so that f is integrable
More informationReal Analysis Notes. Thomas Goller
Real Analysis Notes Thomas Goller September 4, 2011 Contents 1 Abstract Measure Spaces 2 1.1 Basic Definitions........................... 2 1.2 Measurable Functions........................ 2 1.3 Integration..............................
More informationProblem set 5, Real Analysis I, Spring, otherwise. (a) Verify that f is integrable. Solution: Compute since f is even, 1 x (log 1/ x ) 2 dx 1
Problem set 5, Real Analysis I, Spring, 25. (5) Consider the function on R defined by f(x) { x (log / x ) 2 if x /2, otherwise. (a) Verify that f is integrable. Solution: Compute since f is even, R f /2
More informationReal Analysis Qualifying Exam May 14th 2016
Real Analysis Qualifying Exam May 4th 26 Solve 8 out of 2 problems. () Prove the Banach contraction principle: Let T be a mapping from a complete metric space X into itself such that d(tx,ty) apple qd(x,
More informationLEBESGUE INTEGRATION. Introduction
LEBESGUE INTEGATION EYE SJAMAA Supplementary notes Math 414, Spring 25 Introduction The following heuristic argument is at the basis of the denition of the Lebesgue integral. This argument will be imprecise,
More information3. (a) What is a simple function? What is an integrable function? How is f dµ defined? Define it first
Math 632/6321: Theory of Functions of a Real Variable Sample Preinary Exam Questions 1. Let (, M, µ) be a measure space. (a) Prove that if µ() < and if 1 p < q
More informationMath 5051 Measure Theory and Functional Analysis I Homework Assignment 2
Math 551 Measure Theory and Functional nalysis I Homework ssignment 2 Prof. Wickerhauser Due Friday, September 25th, 215 Please do Exercises 1, 4*, 7, 9*, 11, 12, 13, 16, 21*, 26, 28, 31, 32, 33, 36, 37.
More informationReal Analysis, October 1977
Real Analysis, October 1977 1. Prove that the sequence 2, 2 2, 2 2 2,... converges and find its limit. 2. For which real-x does x n lim n 1 + x 2n converge? On which real sets is the convergence uniform?
More informationMATH 409 Advanced Calculus I Lecture 10: Continuity. Properties of continuous functions.
MATH 409 Advanced Calculus I Lecture 10: Continuity. Properties of continuous functions. Continuity Definition. Given a set E R, a function f : E R, and a point c E, the function f is continuous at c if
More informationPrinciple of Mathematical Induction
Advanced Calculus I. Math 451, Fall 2016, Prof. Vershynin Principle of Mathematical Induction 1. Prove that 1 + 2 + + n = 1 n(n + 1) for all n N. 2 2. Prove that 1 2 + 2 2 + + n 2 = 1 n(n + 1)(2n + 1)
More informationEntrance Exam, Real Analysis September 1, 2009 Solve exactly 6 out of the 8 problems. Compute the following and justify your computation: lim
1. Let n be positive integers. ntrnce xm, Rel Anlysis September 1, 29 Solve exctly 6 out of the 8 problems. Sketch the grph of the function f(x): f(x) = lim e x2n. Compute the following nd justify your
More informationSolutions Final Exam May. 14, 2014
Solutions Final Exam May. 14, 2014 1. Determine whether the following statements are true or false. Justify your answer (i.e., prove the claim, derive a contradiction or give a counter-example). (a) (10
More informationAnalysis Comprehensive Exam Questions Fall F(x) = 1 x. f(t)dt. t 1 2. tf 2 (t)dt. and g(t, x) = 2 t. 2 t
Analysis Comprehensive Exam Questions Fall 2. Let f L 2 (, ) be given. (a) Prove that ( x 2 f(t) dt) 2 x x t f(t) 2 dt. (b) Given part (a), prove that F L 2 (, ) 2 f L 2 (, ), where F(x) = x (a) Using
More informationNotions such as convergent sequence and Cauchy sequence make sense for any metric space. Convergent Sequences are Cauchy
Banach Spaces These notes provide an introduction to Banach spaces, which are complete normed vector spaces. For the purposes of these notes, all vector spaces are assumed to be over the real numbers.
More informationNotes on the Lebesgue Integral by Francis J. Narcowich Septemmber, 2014
1 Introduction Notes on the Lebesgue Integral by Francis J. Narcowich Septemmber, 2014 In the definition of the Riemann integral of a function f(x), the x-axis is partitioned and the integral is defined
More informationLogical Connectives and Quantifiers
Chapter 1 Logical Connectives and Quantifiers 1.1 Logical Connectives 1.2 Quantifiers 1.3 Techniques of Proof: I 1.4 Techniques of Proof: II Theorem 1. Let f be a continuous function. If 1 f(x)dx 0, then
More informationDRAFT - Math 101 Lecture Note - Dr. Said Algarni
2 Limits 2.1 The Tangent Problems The word tangent is derived from the Latin word tangens, which means touching. A tangent line to a curve is a line that touches the curve and a secant line is a line that
More informationTools from Lebesgue integration
Tools from Lebesgue integration E.P. van den Ban Fall 2005 Introduction In these notes we describe some of the basic tools from the theory of Lebesgue integration. Definitions and results will be given
More informationvan Rooij, Schikhof: A Second Course on Real Functions
vanrooijschikhof.tex April 25, 2018 van Rooij, Schikhof: A Second Course on Real Functions Notes from [vrs]. Introduction A monotone function is Riemann integrable. A continuous function is Riemann integrable.
More informationAdvanced Calculus Math 127B, Winter 2005 Solutions: Final. nx2 1 + n 2 x, g n(x) = n2 x
. Define f n, g n : [, ] R by f n (x) = Advanced Calculus Math 27B, Winter 25 Solutions: Final nx2 + n 2 x, g n(x) = n2 x 2 + n 2 x. 2 Show that the sequences (f n ), (g n ) converge pointwise on [, ],
More informationMetric Spaces. Exercises Fall 2017 Lecturer: Viveka Erlandsson. Written by M.van den Berg
Metric Spaces Exercises Fall 2017 Lecturer: Viveka Erlandsson Written by M.van den Berg School of Mathematics University of Bristol BS8 1TW Bristol, UK 1 Exercises. 1. Let X be a non-empty set, and suppose
More informationIntegration on Measure Spaces
Chapter 3 Integration on Measure Spaces In this chapter we introduce the general notion of a measure on a space X, define the class of measurable functions, and define the integral, first on a class of
More information1 Solutions to selected problems
1 Solutions to selected problems 1. Let A B R n. Show that int A int B but in general bd A bd B. Solution. Let x int A. Then there is ɛ > 0 such that B ɛ (x) A B. This shows x int B. If A = [0, 1] and
More informationMATH 140B - HW 5 SOLUTIONS
MATH 140B - HW 5 SOLUTIONS Problem 1 (WR Ch 7 #8). If I (x) = { 0 (x 0), 1 (x > 0), if {x n } is a sequence of distinct points of (a,b), and if c n converges, prove that the series f (x) = c n I (x x n
More informationThree hours THE UNIVERSITY OF MANCHESTER. 24th January
Three hours MATH41011 THE UNIVERSITY OF MANCHESTER FOURIER ANALYSIS AND LEBESGUE INTEGRATION 24th January 2013 9.45 12.45 Answer ALL SIX questions in Section A (25 marks in total). Answer THREE of the
More informationChapter 6. Integration. 1. Integrals of Nonnegative Functions. a j µ(e j ) (ca j )µ(e j ) = c X. and ψ =
Chapter 6. Integration 1. Integrals of Nonnegative Functions Let (, S, µ) be a measure space. We denote by L + the set of all measurable functions from to [0, ]. Let φ be a simple function in L +. Suppose
More informationMidterm 1. Every element of the set of functions is continuous
Econ 200 Mathematics for Economists Midterm Question.- Consider the set of functions F C(0, ) dened by { } F = f C(0, ) f(x) = ax b, a A R and b B R That is, F is a subset of the set of continuous functions
More informationFINAL REVIEW FOR MATH The limit. a n. This definition is useful is when evaluating the limits; for instance, to show
FINAL REVIEW FOR MATH 500 SHUANGLIN SHAO. The it Define a n = A: For any ε > 0, there exists N N such that for any n N, a n A < ε. This definition is useful is when evaluating the its; for instance, to
More informationSummary of Real Analysis by Royden
Summary of Real Analysis by Royden Dan Hathaway May 2010 This document is a summary of the theorems and definitions and theorems from Part 1 of the book Real Analysis by Royden. In some areas, such as
More informationProblems for Chapter 3.
Problems for Chapter 3. Let A denote a nonempty set of reals. The complement of A, denoted by A, or A C is the set of all points not in A. We say that belongs to the interior of A, Int A, if there eists
More informationconverges as well if x < 1. 1 x n x n 1 1 = 2 a nx n
Solve the following 6 problems. 1. Prove that if series n=1 a nx n converges for all x such that x < 1, then the series n=1 a n xn 1 x converges as well if x < 1. n For x < 1, x n 0 as n, so there exists
More informationLEBESGUE MEASURE AND L2 SPACE. Contents 1. Measure Spaces 1 2. Lebesgue Integration 2 3. L 2 Space 4 Acknowledgments 9 References 9
LBSGU MASUR AND L2 SPAC. ANNI WANG Abstract. This paper begins with an introduction to measure spaces and the Lebesgue theory of measure and integration. Several important theorems regarding the Lebesgue
More informationMath 140A - Fall Final Exam
Math 140A - Fall 2014 - Final Exam Problem 1. Let {a n } n 1 be an increasing sequence of real numbers. (i) If {a n } has a bounded subsequence, show that {a n } is itself bounded. (ii) If {a n } has a
More informationTHEOREMS, ETC., FOR MATH 515
THEOREMS, ETC., FOR MATH 515 Proposition 1 (=comment on page 17). If A is an algebra, then any finite union or finite intersection of sets in A is also in A. Proposition 2 (=Proposition 1.1). For every
More informationX n D X lim n F n (x) = F (x) for all x C F. lim n F n(u) = F (u) for all u C F. (2)
14:17 11/16/2 TOPIC. Convergence in distribution and related notions. This section studies the notion of the so-called convergence in distribution of real random variables. This is the kind of convergence
More informationSummer Jump-Start Program for Analysis, 2012 Song-Ying Li
Summer Jump-Start Program for Analysis, 01 Song-Ying Li 1 Lecture 6: Uniformly continuity and sequence of functions 1.1 Uniform Continuity Definition 1.1 Let (X, d 1 ) and (Y, d ) are metric spaces and
More informationEconomics 204 Summer/Fall 2011 Lecture 5 Friday July 29, 2011
Economics 204 Summer/Fall 2011 Lecture 5 Friday July 29, 2011 Section 2.6 (cont.) Properties of Real Functions Here we first study properties of functions from R to R, making use of the additional structure
More informationMeasure and Integration: Solutions of CW2
Measure and Integration: s of CW2 Fall 206 [G. Holzegel] December 9, 206 Problem of Sheet 5 a) Left (f n ) and (g n ) be sequences of integrable functions with f n (x) f (x) and g n (x) g (x) for almost
More informationAnalysis/Calculus Review Day 2
Analysis/Calculus Review Day 2 AJ Friend ajfriend@stanford.edu Arvind Saibaba arvindks@stanford.edu Institute of Computational and Mathematical Engineering Stanford University September 20, 2011 Continuity
More informationNotes on the Lebesgue Integral by Francis J. Narcowich November, 2013
Notes on the Lebesgue Integral by Francis J. Narcowich November, 203 Introduction In the definition of the Riemann integral of a function f(x), the x-axis is partitioned and the integral is defined in
More informationMath 321 Final Examination April 1995 Notation used in this exam: N. (1) S N (f,x) = f(t)e int dt e inx.
Math 321 Final Examination April 1995 Notation used in this exam: N 1 π (1) S N (f,x) = f(t)e int dt e inx. 2π n= N π (2) C(X, R) is the space of bounded real-valued functions on the metric space X, equipped
More informationSOME PROBLEMS IN REAL ANALYSIS.
SOME PROBLEMS IN REAL ANALYSIS. Prepared by SULEYMAN ULUSOY PROBLEM 1. (1 points) Suppose f n : X [, ] is measurable for n = 1, 2, 3,...; f 1 f 2 f 3... ; f n (x) f(x) as n, for every x X. a)give a counterexample
More information2 Lebesgue integration
2 Lebesgue integration 1. Let (, A, µ) be a measure space. We will always assume that µ is complete, otherwise we first take its completion. The example to have in mind is the Lebesgue measure on R n,
More informationREAL ANALYSIS I HOMEWORK 4
REAL ANALYSIS I HOMEWORK 4 CİHAN BAHRAN The questions are from Stein and Shakarchi s text, Chapter 2.. Given a collection of sets E, E 2,..., E n, construct another collection E, E 2,..., E N, with N =
More informationL p Functions. Given a measure space (X, µ) and a real number p [1, ), recall that the L p -norm of a measurable function f : X R is defined by
L p Functions Given a measure space (, µ) and a real number p [, ), recall that the L p -norm of a measurable function f : R is defined by f p = ( ) /p f p dµ Note that the L p -norm of a function f may
More informationREAL AND COMPLEX ANALYSIS
REAL AND COMPLE ANALYSIS Third Edition Walter Rudin Professor of Mathematics University of Wisconsin, Madison Version 1.1 No rights reserved. Any part of this work can be reproduced or transmitted in any
More informationContinuous Functions on Metric Spaces
Continuous Functions on Metric Spaces Math 201A, Fall 2016 1 Continuous functions Definition 1. Let (X, d X ) and (Y, d Y ) be metric spaces. A function f : X Y is continuous at a X if for every ɛ > 0
More informationLebesgue Integration on R n
Lebesgue Integration on R n The treatment here is based loosely on that of Jones, Lebesgue Integration on Euclidean Space We give an overview from the perspective of a user of the theory Riemann integration
More informationContinuity. Chapter 4
Chapter 4 Continuity Throughout this chapter D is a nonempty subset of the real numbers. We recall the definition of a function. Definition 4.1. A function from D into R, denoted f : D R, is a subset of
More informationMetric Spaces and Topology
Chapter 2 Metric Spaces and Topology From an engineering perspective, the most important way to construct a topology on a set is to define the topology in terms of a metric on the set. This approach underlies
More informationd(x n, x) d(x n, x nk ) + d(x nk, x) where we chose any fixed k > N
Problem 1. Let f : A R R have the property that for every x A, there exists ɛ > 0 such that f(t) > ɛ if t (x ɛ, x + ɛ) A. If the set A is compact, prove there exists c > 0 such that f(x) > c for all x
More information, but still n=0 b n = b n = 1 n 6 n k=1k 5 a k. sin(x 2 + ax)
Analysis SEP Summer 25 Problem. (a)suppose (a n ) n= is a sequence of nonnegative real numbers such that n= a n
More informationFINAL REVIEW Answers and hints Math 311 Fall 2017
FINAL RVIW Answers and hints Math 3 Fall 7. Let R be a Jordan region and let f : R be integrable. Prove that the graph of f, as a subset of R 3, has zero volume. Let R be a rectangle with R. Since f is
More informationAnalysis Finite and Infinite Sets The Real Numbers The Cantor Set
Analysis Finite and Infinite Sets Definition. An initial segment is {n N n n 0 }. Definition. A finite set can be put into one-to-one correspondence with an initial segment. The empty set is also considered
More information+ 2x sin x. f(b i ) f(a i ) < ɛ. i=1. i=1
Appendix To understand weak derivatives and distributional derivatives in the simplest context of functions of a single variable, we describe without proof some results from real analysis (see [7] and
More informationMATH 5640: Fourier Series
MATH 564: Fourier Series Hung Phan, UMass Lowell September, 8 Power Series A power series in the variable x is a series of the form a + a x + a x + = where the coefficients a, a,... are real or complex
More informationProblem set 1, Real Analysis I, Spring, 2015.
Problem set 1, Real Analysis I, Spring, 015. (1) Let f n : D R be a sequence of functions with domain D R n. Recall that f n f uniformly if and only if for all ɛ > 0, there is an N = N(ɛ) so that if n
More informationABSTRACT INTEGRATION CHAPTER ONE
CHAPTER ONE ABSTRACT INTEGRATION Version 1.1 No rights reserved. Any part of this work can be reproduced or transmitted in any form or by any means. Suggestions and errors are invited and can be mailed
More informationSummer Jump-Start Program for Analysis, 2012 Song-Ying Li. 1 Lecture 7: Equicontinuity and Series of functions
Summer Jump-Start Program for Analysis, 0 Song-Ying Li Lecture 7: Equicontinuity and Series of functions. Equicontinuity Definition. Let (X, d) be a metric space, K X and K is a compact subset of X. C(K)
More informationMath 172 HW 1 Solutions
Math 172 HW 1 Solutions Joey Zou April 15, 2017 Problem 1: Prove that the Cantor set C constructed in the text is totally disconnected and perfect. In other words, given two distinct points x, y C, there
More informationFUNDAMENTALS OF REAL ANALYSIS by. II.1. Prelude. Recall that the Riemann integral of a real-valued function f on an interval [a, b] is defined as
FUNDAMENTALS OF REAL ANALYSIS by Doğan Çömez II. MEASURES AND MEASURE SPACES II.1. Prelude Recall that the Riemann integral of a real-valued function f on an interval [a, b] is defined as b n f(xdx :=
More informationReview of measure theory
209: Honors nalysis in R n Review of measure theory 1 Outer measure, measure, measurable sets Definition 1 Let X be a set. nonempty family R of subsets of X is a ring if, B R B R and, B R B R hold. bove,
More informationSome analysis problems 1. x x 2 +yn2, y > 0. g(y) := lim
Some analysis problems. Let f be a continuous function on R and let for n =,2,..., F n (x) = x (x t) n f(t)dt. Prove that F n is n times differentiable, and prove a simple formula for its n-th derivative.
More informationMcGill University Math 354: Honors Analysis 3
Practice problems McGill University Math 354: Honors Analysis 3 not for credit Problem 1. Determine whether the family of F = {f n } functions f n (x) = x n is uniformly equicontinuous. 1st Solution: The
More informationSOLUTIONS OF SELECTED PROBLEMS
SOLUTIONS OF SELECTED PROBLEMS Problem 36, p. 63 If µ(e n < and χ En f in L, then f is a.e. equal to a characteristic function of a measurable set. Solution: By Corollary.3, there esists a subsequence
More informationImmerse Metric Space Homework
Immerse Metric Space Homework (Exercises -2). In R n, define d(x, y) = x y +... + x n y n. Show that d is a metric that induces the usual topology. Sketch the basis elements when n = 2. Solution: Steps
More informationvan Rooij, Schikhof: A Second Course on Real Functions
vanrooijschikhofproblems.tex December 5, 2017 http://thales.doa.fmph.uniba.sk/sleziak/texty/rozne/pozn/books/ van Rooij, Schikhof: A Second Course on Real Functions Some notes made when reading [vrs].
More informationIowa State University. Instructor: Alex Roitershtein Summer Homework #5. Solutions
Math 50 Iowa State University Introduction to Real Analysis Department of Mathematics Instructor: Alex Roitershtein Summer 205 Homework #5 Solutions. Let α and c be real numbers, c > 0, and f is defined
More information6.2 Fubini s Theorem. (µ ν)(c) = f C (x) dµ(x). (6.2) Proof. Note that (X Y, A B, µ ν) must be σ-finite as well, so that.
6.2 Fubini s Theorem Theorem 6.2.1. (Fubini s theorem - first form) Let (, A, µ) and (, B, ν) be complete σ-finite measure spaces. Let C = A B. Then for each µ ν- measurable set C C the section x C is
More informationAnalysis Comprehensive Exam Questions Fall 2008
Analysis Comprehensive xam Questions Fall 28. (a) Let R be measurable with finite Lebesgue measure. Suppose that {f n } n N is a bounded sequence in L 2 () and there exists a function f such that f n (x)
More informationMath212a1413 The Lebesgue integral.
Math212a1413 The Lebesgue integral. October 28, 2014 Simple functions. In what follows, (X, F, m) is a space with a σ-field of sets, and m a measure on F. The purpose of today s lecture is to develop the
More informationconvergence theorem in abstract set up. Our proof produces a positive integrable function required unlike other known
https://sites.google.com/site/anilpedgaonkar/ profanilp@gmail.com 218 Chapter 5 Convergence and Integration In this chapter we obtain convergence theorems. Convergence theorems will apply to various types
More information3 hours UNIVERSITY OF MANCHESTER. 22nd May and. Electronic calculators may be used, provided that they cannot store text.
3 hours MATH40512 UNIVERSITY OF MANCHESTER DYNAMICAL SYSTEMS AND ERGODIC THEORY 22nd May 2007 9.45 12.45 Answer ALL four questions in SECTION A (40 marks in total) and THREE of the four questions in SECTION
More informationMATH MEASURE THEORY AND FOURIER ANALYSIS. Contents
MATH 3969 - MEASURE THEORY AND FOURIER ANALYSIS ANDREW TULLOCH Contents 1. Measure Theory 2 1.1. Properties of Measures 3 1.2. Constructing σ-algebras and measures 3 1.3. Properties of the Lebesgue measure
More informationContinuity. Chapter 4
Chapter 4 Continuity Throughout this chapter D is a nonempty subset of the real numbers. We recall the definition of a function. Definition 4.1. A function from D into R, denoted f : D R, is a subset of
More informationMAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9
MAT 570 REAL ANALYSIS LECTURE NOTES PROFESSOR: JOHN QUIGG SEMESTER: FALL 204 Contents. Sets 2 2. Functions 5 3. Countability 7 4. Axiom of choice 8 5. Equivalence relations 9 6. Real numbers 9 7. Extended
More informationL p Spaces and Convexity
L p Spaces and Convexity These notes largely follow the treatments in Royden, Real Analysis, and Rudin, Real & Complex Analysis. 1. Convex functions Let I R be an interval. For I open, we say a function
More informationReal Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi
Real Analysis Math 3AH Rudin, Chapter # Dominique Abdi.. If r is rational (r 0) and x is irrational, prove that r + x and rx are irrational. Solution. Assume the contrary, that r+x and rx are rational.
More informationReal Analysis Comprehensive Exam Fall A(k, ε) is of Lebesgue measure zero.
Real Analysis Comprehensive Exam Fall 2002 by XYC Good luck! [1] For ε>0andk>0, denote by A(k, ε) thesetofx Rsuch that x p q 1 for any integers p, q with q 0. k q 2+ε Show that R \ k=1 A(k, ε) is of Lebesgue
More informationProblem List MATH 5143 Fall, 2013
Problem List MATH 5143 Fall, 2013 On any problem you may use the result of any previous problem (even if you were not able to do it) and any information given in class up to the moment the problem was
More informationAdvanced Analysis Qualifying Examination Department of Mathematics and Statistics University of Massachusetts. Tuesday, January 16th, 2018
NAME: Advanced Analysis Qualifying Examination Department of Mathematics and Statistics University of Massachusetts Tuesday, January 16th, 2018 Instructions 1. This exam consists of eight (8) problems
More informationMATH 5616H INTRODUCTION TO ANALYSIS II SAMPLE FINAL EXAM: SOLUTIONS
MATH 5616H INTRODUCTION TO ANALYSIS II SAMPLE FINAL EXAM: SOLUTIONS You may not use notes, books, etc. Only the exam paper, a pencil or pen may be kept on your desk during the test. Calculators are not
More informationThe reference [Ho17] refers to the course lecture notes by Ilkka Holopainen.
Department of Mathematics and Statistics Real Analysis I, Fall 207 Solutions to Exercise 6 (6 pages) riikka.schroderus at helsinki.fi Note. The course can be passed by an exam. The first possible exam
More information