Math 73/103 Final Exam
Instructions: You should return your exam to me in my oce between 2:00 and 3:00 on
Monday afternoon, December 5, 2011. This is a closed book, closed notes exam.
1. (20) Recall that the complex conjugate of z = x + iy is given by z :
Homework for Math 103
Assignment One Due September 30
1. Show that the countable union of sets of measure zero in R has measure zero.
ANS: Suppose that En has measure zero for n = 1, 2, . . . , and let E = En . Let > 0. By assumption, there are
intervals
Second Homework Assignment
Math 73/103
Due Wednesday, October 19th
1. Page 32 of the text, problem #6. (Note that we have already shown that M is a -algebra
so there is no need to show it again.)
ANS: We already know M is a -algebra. Let cfw_Ei be a coun
How Many Borel Sets are There?
Object. This series of exercises is designed to lead to the conclusion that if BR is the algebra of Borel sets in R, then
Card(BR ) = c := Card(R).
This is the conclusion of problem 4. As a bonus, we also get some insight in
Math 73/103 Assignment Three
Due Friday, November 4th
Clarification: Since at least one person found some legitimate ambiguities in their notes,
let me be clear about our terminology. Lebesgue measure, (R, M, m), is the complete measure coming from the ex
Theorem 1 (Folland Theorem 2.28). Suppose that f is a bounded real-valued
function on [a, b].
1. If f is Riemann integrable, then f is Lebesgue measurable (and therefore
integrable). Furthermore
b
R
f=
f (x) dm(x).
(1)
[a,b]
a
(Henceforth, we will dispens
Math 73/103 Midterm
1. (15) Give precise statements (no proofs necessary on this problem) of Littlewoods Three
Principles:
(a) Every Lebesgue measurable set is almost a disjoint union of intervals.
ANS: If m(E ) < and > 0, then there is a nite set of disj
Math 73/103: Homework on the Cauchy-Riemann Equations
Due TBA
1. Suppose that is a region in C, and that f H (). Show that if f (z ) = 0 for all z ,
then f is constant.
Let be a domain in C and assume that f : C is a function. Of course, we can
view as an
Math 73/103: Homework on the Cauchy-Riemann
Equations
Problems 16 are due Monday, November 21, 2011.
Problems 711 are just for your edication. (But just might appear on some sort of
nal exam.)
The remaining problems are due the last day of class, Monda
Homework for Math 103
Assignment One Due September 30
1. Show that the countable union of sets of measure zero in R has measure zero.
2. Suppose f : [a, b] R is bounded, and let P and Q be partitions of [a, b]. Prove that L(f, P ) U (f, Q),
where L(f, P )