EXAMINATION PAPER Examination Session: May Year: 2018

 Exam Code: MATH4201-WE01 

 Title: Analysis IV Time Allowed: 3 hours Additional Material provided: None Materials Permitted: None Calculators Permitted: No Models Permitted: Use of electronic calculators is forbidden. Visiting Students may use dictionaries: No Instructions to Candidates: Credit will be given for: the best TWO answers from Section A, the best THREE answers from Section B, AND the answer to the question in Section C. Questions in Section B and C carry TWICE as many marks as those in Section A. Revision: ED01/2018 University of Durham Copyright ED01/2018 CONTINUED Page number Exam code MATH4201-WE01 1. (a) (b) LetQ[ 2]={a+b 2|a,bQ}R. ShowthatQ[ 2]iscountable. (c) Show that the set of all infinite sequences of natural numbers is uncountable. 2. Let {an} be a sequence of real numbers, and let A be the set of all elements of {an}. 2 of 3 Define what means for a set to be countable. SECTION A (a) Define lim sup an. (b) Show that sup A lim sup an. (c) Assume that lim sup an A. Does this imply that sup A A? 3. (a) Define what it means for a set E R to be measurable. (b) Show that if E R is measurable and bounded and > 0, then there exists a finite collection {E1, . . . , En} of mutually disjoint measurable sets such that ni=1Ei =Eandm(Ei)foralli=1,...,n. (c) Is the assertion of (b) true if E is unbounded of finite measure? Define what means for f : R R to be measurable. (b) Prove that f(x) := sin(3x) is measurable. You may use that open intervals are measurable. (c) Let f : R R be measurable, a R and define the function g : R R by 4. (a) g(x) := f(x a). Prove that g is measurable. Define f for a nonnegative measurable function f : R R. 5. (a) (b) Prove that for measurable functions f,g with 0 f(x) g(x) we have f g. (c) Letf beasin(a),assumethatf =0,c>0,andAc :={xR|f(x)>c}. Prove that m(Ac) = 0, where m denotes the Lebesgue measure of R. State the Lemma of Fatou. 0, xn; fn(x)= 1, x>n. 6. (a) (b) Let Do the assumptions of the Lemma of Fatou apply to the sequence? Prove your answer. (c) Does the conclusion of the Lemma of Fatou apply to the sequence fn as in (b)? Prove your answer. University of Durham Copyright ED01/2018 END Page number 3 of 3 Exam code MATH4201-WE01 SECTION B Define the outer measure m(E) of a set E R. m(B) for A B. (c) Show that a set E R is measurable if and only if for every > 0 there exist anopensetU andaclosedsetF suchthatF EU andm(U\F)<. (d) Let E R have finite outer measure. Show that there exists a countable intersection G of open sets such that E G and m(E) = m(G). Does this imply that m(G \ E) = 0? 8. A set A R is called nowhere dense if every open non-empty set U R has an open non-empty subset U0 U such that U0 A = . A set B R is called dense if the closure of B coincides with R. (a) Let N N. Show that the set {pq |p,q N, p N} is nowhere dense. (b) Let B R be dense. Is it true that R \ B has to be nowhere dense? (c) LetARbenowheredense. IsittruethatR\Ahastobedense? (d) Show that for every > 0 there exists a nowhere dense set E R such that m(R \ E) < . 7. (a) (b) Use the definition (a) to show that the outer measure is monotone, i.e. m(A) 9. (a) Define the collection of functions L1([0,2]) and the collection of functions L2([0, 2]). (b) Prove or disprove by counterexample the following claim : L1([0, 2]) L2([0, 2]). (c) State the Dominated Convergence Theorem. Define an inner product on L2([0, 2]) and define what it means that L2([0, 2]) is a Hilbert space. 10. (a) (b) State and prove the Bessel Inequality on the Hilbert space L2([0,2]). (c) Define functions fn : R R by fn(x) = (1/n)[0,n] and f(x) = 0. Here, [0,n] has value 1 on [0, n] and vanishes elsewhere. Show that fn converges uniformly to f but that lim fn = f. Why does this not contradict the Monotone Convergence Theorem ? SECTION C n (c) Show that there exists a pointwise limit f(x) = limn fn(x). Find explicitly a closed set F [/2, /2] such that m([/2, /2] \ F ) < 1/5 and the convergence of {fn} to f on F is uniform. (d) Give an example of a sequence of measurable functions gn : R R which converges pointwise to a function g : R R and to which the conclusion of the Egoroffs Theorem does not apply. State Egoroffs Theorem. for x [/2, /2] Q. Show that all fn are measurable. 11. (a) (b) Let fn : [/2,/2] R, fn(x) = tanx for x (/2,/2)(R\Q), fn(x) = 0 University of Durham Copyright