MTH 535 Fall 06 -- Tips for Exam 2 Exam 2 is scheduled for Friday, December 8 , 3-5, Tyler 106. Exam 2 covers classes 11-23 and homework assignments 6-9. Notes and videos for class 23 that we missed are posted. Please study the material, become familiar with the Lebesgue Dominated Convergence and its proof. This theorem is the main part of class 23 and this is what you need for Homework 9, problems 5, 6. As for every exam, for Exam 2 you are expected to know and be able to state all definitions, propositions, and theorems given in class and in homework assignments as well as their proofs. You should know all of the important examples studied in class. You are expected to know solutions to all homework problems. Some problems on the exam will ask you to state and prove a theorem proved in class; some will ask you to state a definiton and give an example; some will be similar to homework problems. While I was browsing my notes and homework assignments, the following questions stood out as good candidates for exam questions. (I am indicating below theorems, propositions, examples studied in class without stating precisely all the assumptions. Consult your notes for that. I am using numbers as we did in class, but of course, on the exam I will not expect you to remember numbers of defintions, propositions, theorem etc.) Items below are listed in no particular order. Some overlap with the homework problems listed below. -- State the Lebesgue Dominated Convergence Theorem. Prove the Theorem. -- State Fatou's Lemma. Give an example that the inequalty in the Lemma might be strict. -- State the Monotone Convergence Theoerem. Use the Theorem to show that the Riemann improper integral of 1/x^2 in [1,infinity) is equal to the Lebesgue integral of 1/x^2 in [1,infinity). -- Define the positve and the negative parts of a function f. Show that if a function is measurable, its positve and negative parts are measurable. -- Define the Lebesgue integral and the Lebesgue integrability of a measurable function f on a measurable set E. -- Show that if f is integrable on E then the set of points where f is plus infinity and the set of points where f is minus infinity both have measure 0. -- State Fatou's Lemma. State the Monotone Convergence Theorem. Prove the Monotone Convergence Theorem. -- State the Bounded Convergence Theorem. Prove the Theorem. -- Prove Prop 17.1. -- Prove Prop. 17.2. -- State the Egoroff Theorem. Use the theorem to prove H7 #6. Give an example that the theorem does not hold without the boundedness of D assumption. -- Show that the infimum of a countable collection of measurable functions is measurable. Show that the infimum of an arbitrary collection of measurable functions may not be measurable. -- Show that the limit inferior of a sequence of measurable functions is measurable. -- State the definition of a measurable function. Give an example of a function that is not measurable. Prove Prop. 15.1. -- State and prove Th. 13.1. From homework problems, the following seem particularly good: H6: #2,#3,#4 H7: #1, #2, #3,#5,#6 H8: #2,#3,#4 H9: All. Consider it a practice exam (Of course, the actual exam will be much shorter). Some of the problems listed above will appear, some will not. A problem or two not listed above may appear as well. Who knows? I hope you all get As!