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# FUND CONCEPTS ANLYS MATH 426

UW

GPA 3.76

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This 2 page Class Notes was uploaded by Addison Beer on Wednesday September 9, 2015. The Class Notes belongs to MATH 426 at University of Washington taught by Tatiana Toro in Fall. Since its upload, it has received 19 views. For similar materials see /class/192080/math-426-university-of-washington in Mathematics (M) at University of Washington.

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Date Created: 09/09/15

Math 426576 FINAL PREPARATION Spring 2005 The nal will be on Wednesday June 8 230 420 pm in the usual room LOW 106 It will be comprehensive There will be theoretical questions and short problems on all topics that we covered and some medium length problems on selected topics with at least one on the last topic Chapter 10 You can bring two two sided sheet of notes or four one sided sheets handwritten only No electronic devices are allowed There will be an optional review on Tuesday June 7 500 600 in C 36 PDL There will be of ce hours on Tuesday June 7 11 am 7 1200 in my of ce as well Your TA will also have of ce hours TBA H 3 9 7 U 03 5 00 p H 0 H H H 3 Theoretical questions for review What is an algebra of sets What is a U algebra What is a monotone class Give examples State the Monotone Class Lemma What is a measure on an algebra of sets on a U algebra State the Caratheodory extension Theorem in the general setting Can you remember which statements required the measure to be nite U nite Know examples which show why these assumptions are necessary Explain the construction of a product U algebra and the product measure Explain the difference between the Lebesgue measure in R2 and the product of Lebesgue measures on State the Tonelli and Fubini Theorems What is the difference between them Explain the connection between nite Borel measures on the real line and the Lebesgue Stieltjes measures Give a de nition of an absolutely continuous continuous and singular measure with respect to the Lebesgue measure on the real line Give examples of each Construct an uncountable family of pairwise mutually singular Borel measures on the real line State the Radon Nikodym and Lebesgue decomposition theorems Prove that the Riemann integral agrees with the Lebesgue integral for a continuous functions on an interval of nite length b for continuous non negative functions on the real line State the Monotone Convergence Theorem Fatou7s Lemma and the Lebesgue Dominated Convergence Theorem Give examples of how these theorems apply and what can go wrong if some of the assumptions are not satis ed 13 State Egoro 7s Theorem Know the de nitions of convergence in LP in measure uniform almost uniform and almost everywhere Know the standard implications and examples showing what fails eg convergence in measure does not imply ae convergence etc 14 De nition of LF spaces including Loo don7t forget about equivalence classesl State the Holders and Minkowski7s inequalities lnterpret them in the case of counting measure In general be able to interpret all our theorems in the case of counting measurel State the Completeness Theorem with all the de nitions H CH State the theorems about differentiation under the integral sign and limits of integrals depending on parameter 16 Review theoretical questions for the 1st midterm Practice problems for the nal 1 Suppose M is complete measure a If f measurable and g f ae then 9 is measurable b If M is not complete then a is in general false 2 Let S be an in nite U algebra on a set X a Show that S contains an in nite sequence of disjoint nonempty sets b Show that S is uncountable thus there are no countably in nite U algebrasl 3 Let X be the set of integers and let X be the U algebra of all subsets of X If A is in nite let MA 00 Otherwise let MA 0 ls M a measure on X X 4 Give an example of two measures M and V on a measurable space X X such that neither VltltMnorMltltVnorV1M 5 Let A be a Lebesgue measurable subset of R Prove that the function f A7oo H A is continuous where A is the Lebesgue measure 6 Consider R with the Borel U algebra and the Lebesgue measure A Let fn 1nXnoo Find a function such that fn converges almost everywhere to f and show that fdAJLrnOfndA Why does this not contradict the MCT Why does this not contradict the LDCT Why does this not contradict the Fatou7s Lemma 7 In each of the following cases give explicit examples of a sequence of measurable functions on a carefully speci ed measurable space which show that a ae convergence does not imply the convergence in LP b convergence in Lp does not imply ae convergence c convergence in measure does not imply ae convergence 815 From Bartle 10G10H10K10N10010P10Q10R

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