Ma 4121: Introduction to Lebesgue Integration
Solutions to Homework Assignment 1
Prof. Wickerhauser
Due Thursday, January 31st, 2013
Please return your solutions to the instructor by the end of class on the due date. You may collaborate
on these problems
Answers to Homework 6, Math 4121
equicontinuous=uniformly equicontinuous, for brevity in the following.
(1) If cfw_fn is an equicontinuous sequence of functions on a compact
interval and fn f pointwise, prove that the convergence is
uniform.
First, we pr
Ma 4121: Introduction to Lebesgue Integration
Solutions to Homework Assignment 3
Prof. Wickerhauser
Due Thursday, February 28th, 2013
Please return your solutions to the instructor by the end of class on the due date. You may collaborate
on these problems
1. -algebras
Denition 1. Let X be any set and let F be a collection of subsets of
X. We say that F is a -algebra (on X), if it satises the following.
(1) X F.
(2) If A F, then Ac F.
(3) If A1 , A2 , F, a countable collection, then An F.
n=1
In the above s
Answers to Homework 10, Math 4121
x
(1) Prove that x log 2 L1 ([0, 1]).
1+x
The integrand is dominated by |x log x| = x log x. x log x
L1 ([0, 1]) can be seen by integrating by parts.
(2) Prove that 1+x41sin2 x L1 ([0, ).
Consider A = cfw_x [0, )| sin2 x
Midterm, Math 4121, March 20, 2014
(1) (a) Dene lim sup an for a sequence cfw_an of real numbers.
(b) If cfw_an , cfw_bn are two sequences of real numbers, prove that
lim sup(an + bn ) lim sup an + lim sup bn .
Let A = lim sup an , B = lim sup bn . Give
Answers to Homework 11, Math 4121
(1) For 0 < r < p < s < , prove that Lr Ls Lp . Further, if
(X) < , prove that Ls Lr if 0 < r < s < .
Writing p = (1 t)r + ts with t (0, 1) and using the fact
that (x) = ax is a convex function for any a > 0 on R1 , we se
Answers to Homework 12, Math 4121
All problems are routine and should require little or no thinking.
We will denote by n an open set in Rn .
(1) Let f : n R be a function which has all partial derivatives
D1 f, . . . , Dn f for all points in n . If p n wh
Answers to Homework 8
In the following, (X, F, ) will be a measure space.
(1) Let f1 f2 f 0 be a sequence of measurable functions with range [0, ] and lim fn = f . Assume that f1 L1 ().
Then prove that lim X fn d = X f d. Give an example to
show that the
Homework 9, Math 4121, due 3, April 2014
In the following, (X, F, ) will be a measure space. As usual problem(s)
in blue are not to be submitted.
(1) Let denote the Lebesgue measure on [0, 1]. Given any r,
0 < r < 1, show that there is a dense open set E
Homework 7, Math 4121, due 6, March 2014
In the following, (X, F) will be a measurable space.
(1) Let f : X R (or the extended real line) be a function. Prove
that it is measurable if and only if for any rational number q,
cfw_x X|f (x) q F.
We have seen
Answers to Homework 5, Math 4121
(1) (a) Let f be continuous on [a, b], a compact interval. Show
that there exists a polynomial P of degree at most one
such that (f + P )(a) = (f + P )(b) = 0. Conclude that
f can be extended to a continuous function on al
Answers to Homework 4, Math 4121
(1) Prove that (n + 1)z n has radius of convergence 1 and thus
n=0
denes a function F (z) on B(0, 1) with F (0) = 1. Find a power
series expansion for 1/F (z) near the origin.
1
The radius of convergence follows from the f
Midterm 0, Feb 13, 2014
(1) (a) Dene convergence of a sequence cfw_an .
1
(b) Prove that limn n n = 1.
There are many proofs. Let me give an algebraic proof
here, since I have indicated a proof using LHopitals rule
in class.
1
Since for any n N, n n 1, we
Homework 2, Math 4121, due 30 Jan 2014
(1) If cfw_an is a decreasing sequence and an converges, prove that
1
limn nan = 0. This gives another proof that
diverges.
n
Since
an converges, lim an = 0. Since cfw_an is a decreasing
sequence, we see that an 0
Answers to Homework 3, Math 4121
(1) Prove that if fn f, gn g uniformly on a set S, so does
fn +gn f +g. Give an example where fn gn does not converge
uniformly to f g.
Given > 0, we can nd an N so that for all n > N , |fn (x)
f (x)| < , |gn (x) g(x)| <
Answers to Homework 1, Math 4121
(1) Find the lim sup of the sequence cfw_cos n. (Hint: You may use
the fact that the set cfw_a + b|a, b Z is dense in R.)
lim sup cos n = 1. First, it is clear that for any number a > 1,
we have cos n < a. Next we must sho
Ma 4121: Introduction to Lebesgue Integration
Solutions to [Revised] Homework Assignment 2
Prof. Wickerhauser
Due Thursday, February 14th, 2013
Please return your solutions to the instructor by the end of class on the due date. You may collaborate
on thes
Ma 4121: Introduction to Lebesgue Integration
Solutions to Homework Assignment 4
Prof. Wickerhauser
Due Thursday, March 28th, 2013
Please return your solutions to the instructor by the end of class on the due date. You may
collaborate on these problems bu
Ma 4121: Introduction to Lebesgue Integration
Solutions to Homework Assignment 6
Prof. Wickerhauser
Due Thursday, April 25th, 2013
Please return your solutions to the instructor by the end of class on the due date. You may collaborate
on these problems bu
Ma 4121: Introduction to Lebesgue Integration
Solutions to Homework Assignment 5
Prof. Wickerhauser
Due Thursday, April 11th, 2013
Please return your solutions to the instructor by the end of class on the due date. You may
collaborate on these problems bu