Final Exam, Math 245C, June 4, 2007. Exams are due June 13 at 6:00 PM in my mailbox. You may use your course notes or any book available, including Folland and Hrmander, but you should write out a o complete solution of each problem. J. Garnett
1. A topol
9. S IGNED MEASURES AND THE R ADON -N IKODYM THEOREM
The purpose of this section is to prove the Radon-Nikodym theorem and discuss generally the
subject of absolutely continuous and singular measures. These will naturally lead to the concept
of signed m
Math 245A, Real Analysis
There are five problems with a total of 50 points.
Problem 1: Let (X, A) be a measurable space and fn : X C for n N be
measurable functions. Consider the set E of all points x X for which the limit
lim fn (x)
245C notes, Spring 2015
Regular meeting on 9 am on Thursdays.
Thm. (Riesz-Thorin) Let (X, F, ) be -finite measure space, Lp be the
space of measurable f : X C with kf kp = ( |f |p d) p < . Pick
p0 , p1 , q0 , q1 [1, ]