Math 523a - Solutions to Final - Fall 2012 1. Let xn be a sequence in R. Define
f (x) =
n=1
exp(-n2 |x - xn |)
Prove that f (x) < a.e. Solution: We will show that f (x) dx < . This implies that f (x) is finite a.e. Applying the monotone converge theorem t
Math 523a - Homework 3 - selected solutions 2. Problem 19, p. 32 in Folland First suppose E is measurable. Then (E) = (E). Since (X) < , (E) = (X) - (E c ) = (X) - (E c ) = (X \ E c ) = (E) So (E) = (E). Now suppose (E) = (E). By part (a) of previous prob
Math 523a - Midterm - In class part 1. Let (X, M, ) be a measure space. Let f L1 (X, M, ) (a) Show that if there is a constant (X) < . > 0 such that f a.e., then
(b) Show that if f > 0 a.e., then is -finite. (Recall that -finite means that there is a coun
Math 523a - Midterm - Take home part solutions 1. Recall that for two sets E, F , we define EF = (E \ F ) (F \ E). And for a subset E of R we define E + x = cfw_y + x : y E. Let m be Lebesgue measure on the real line. Let E R be a Lebesgue measurable set
Math 523a - Midterm - Take home part Due Monday, Oct 22 at 10 am. Rules: You may not talk to anyone about the exam. If the problem is unclear you can ask me for clarification, but I will not give hints. You may refer to your class notes and Folland. But y
Final Exam
Math 523a Real Analysis Prof. Venkataramani This was a three hour exam and the students had to do any six of the eight questions. (1) fn : (0, 1) R is a sequence of continuous functions. Show that the set of points x in (0, 1) such that fn (x)