Homework for Math 103
Assignment One Due September 24
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) a
Second Homework Assignment
Math 73/103
Due Wednesday, October 8, 2014.
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
Math 73/103 Assignment Three
Due 24 October 2014
Clarification: Lets review of notation and terminology. Lebesgue measure, (R, M, m),
is the complete measure coming from the explicit outer measure m we dened in lecture. In
particular, M is the -algebra of
Second Homework Assignment
Math 73/103
Due Wednesday, October 8, 2014.
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.)
2. Page 32 of the text, problem #7.
3. Page 32 of the t
Homework for Math 103
Assignment One Due September 24
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
Math 73/103 Assignment Five
Due Monday, November 17, 2014
1. Note that every nonzero complex number z can be written in polar form: z = rei where
r = |z| and is called an argument of z. Of course the argument of z is only dened up
to a multiple of 2.
(a)
Math 73/103 Assignment Four
Due Date TBA
1. Let be a complex measure on (X, M).
(a) Show that there is a measure and a measurable function : X C so that | = 1,
and such that for all E M,
(E) =
d.
()
E
(Hint: write = 1 2 + i(3 4 ) for measures i . Put 0 =
Math 73/103 Final Exam
Instructions: You should return your exam to me in my oce before noon on Monday,
November 24, 2014.
(a) You must work alone. You may use the text, class notes and past homeworks, but no
other sources allowed.
1
(b) Please turn in yo
Math 73/103 Final Exam
Instructions: You should return your exam to me in my oce before noon on Monday,
November 24, 2014.
(a) You must work alone. You may use the text, class notes and past homeworks, but no
other sources allowed.
1
(b) Please turn in yo
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
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 Five
Due Monday, November 17, 2014
1. Note that every nonzero complex number z can be written in polar form: z = rei where
r = |z| and is called an argument of z. Of course the argument of z is only dened up
to a multiple of 2.
(a)
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 Assignment Three
Due Date TBA
Clarification: Lets review of notation and terminology. Lebesgue measure, (R, M, m),
is the complete measure coming from the explicit outer measure m we dened in lecture. In
particular, M is the -algebra of all m