Math 318 HW #6 Solutions
1. (a) Exercise 16.7. Prove Corollary 10.10, which says that if A1 A2 A3 are measurable subsets of E , then Ai is measurable and m ( Ai ) = limi m (Ai ).
i=1
i=1
Proof. Let Bi = E \Ai for each i. Then
B1 B2 B3
i=1 Bi
and so, by C
Math 318 HW #7
Due 5:00 PM Thursday, March 24
Reading:
Wilcox & Myers 1719.
Problems:
1. Exercise 20.12.
2. Exercise 20.16.
3. We say that a sequence (fn ) dened on a measurable set A R converges in measure to a
function f : A R if
lim mcfw_x A : |fn (x)
Math 318 HW #7 Solutions
1. Exercise 20.12. Let C be the Cantor set. Let D [0, 1] be a nowhere dense measurable set
with m(D) > 0. Then there is a non-measurable set B D. At each stage of the construction
of C and of D, a certain nite number of open inter
Math 318 HW #8
Due 5:00 PM Thursday, April 7
Reading:
Wilcox & Myers 2127.
Problems:
1. (a) Prove Chebyshevs inequality, which says that if f is nonnegative and measurable on the
bounded, measurable set A, then
mcfw_x A : f (x) c
(b) Show that if
A |f |d
Math 318 HW #8 Solutions
1. (a) Prove Chebyshevs inequality, which says that if f is nonnegative and measurable on the
bounded, measurable set A, then
mcfw_x A : f (x) c
1
c
f dm.
A
Proof. Let c R and dene
Ac := cfw_x A : f (x) c,
which is measurable sin
Math 318 HW #9
Due 5:00 PM Thursday, April 14
Reading:
Wilcox & Myers 2830.
Problems:
1. Let A be a bounded, measurable set and let (fn ) be a sequence of measurable functions on A
converging to f . Suppose L(A) and that |fn (x)| (x) for all x A and all n
Math 318 HW #9 Solutions
1. Let A be a bounded, measurable set and let (fn ) be a sequence of measurable functions on A
converging to f . Suppose L(A) and that |fn (x)| (x) for all x A and all n = 1, 2, . . .
Show that
f g dm
fn g dm =
lim
n A
A
if g is m
Math 318 HW #10
Due 3:00 PM Friday, April 29
Reading:
Wilcox & Myers 3237.
Problems:
1. Exercise 34.36.
Comment: The unit ball not being compact is a common feature of innite-dimensional
normed linear spaces which can be a serious nuisance. In fact, one c
Math 318 HW #6
Due 5:00 PM Thursday, March 17
Reading:
Wilcox & Myers 1315.
Problems:
1. (a) Exercise 16.7.
(b) Exercise 16.36.
2. Exercise 16.23.
3. (a) Find a trivial proof of Theorem 12.9 which illustrates why this theorem is not very useful
as stated.
Math 318 HW #5 Solutions
1. Show that the sets Ex dened as part of showing that m is not countably additive (cf.
Example 7.7) form equivalence classes. Use this to conclude that for any x, y [0, 1], either
Ex = Ey or Ex Ey = .
Proof. We want to show that
Math 318 HW #1 Solutions
1. Exercise 6.2.3. Consider the sequence of functions
x
hn (x) =
1 + xn
over the domain [0, ).
(a) Find the pointwise limit of (hn ) on [0, ).
Answer. Let h be the function
x
h(x) = 1/2
0
if 0 x < 1
if x = 1
if x > 1.
Then the cla
Math 318 HW #2 Solutions
1. Exercise 6.2.16. For each n N, let fn be a function dened on [0, 1]. If (fn ) is bounded on
[0, 1]that is, there exists an M > 0 such that |fn (x)| M for all n N and x [0, 1]and
if the collection of functions (fn ) is equiconti
Math 318 HW #3
Due 5:00 PM Thursday, February 10
Reading:
Abbott 6.56.6.
Problems:
1. Exercise 6.5.2.
2. (a) Exercise 2.7.12
(b) Use part (a) to prove Abels Lemma (this is very similar to Exercise 2.7.14(b).
3. Exercise 6.5.8.
4. Exercise 6.5.9.
5. Exerci
Math 318 HW #3 Solutions
1. Exercise 6.5.2. Find suitable coecients (an ) so that the resulting power series
an xn
(a) converges absolutely for all x [1, 1] and diverges o of this set;
Answer. Consider the series
xn
.
n2
n=1
xn
n2
1
since
converges, the W
Math 318 HW #4 Solutions
1. Abbott Exercise 7.4.6. Review the discussion immediately preceding Theorem 7.4.4.
(a) Produce an example of a sequence fn 0 pointwise on [0, 1] where limn
not exist.
Answer. Consider the function fn whose graph is given in Figu
Math 318 HW #5
Due 5:00 PM Thursday, March 3
Reading:
Wilcox & Myers 1012.
Problems:
1. Show that the sets Ex dened as part of showing that m is not countably additive (cf.
Example 7.7) form equivalence classes. Use this to conclude that for any x, y [0,
Analysis II (Math 318)
Spring 2011
1
Technicalities
Instructor: Clay Shonkwiler (cshonkwi@haverford.edu)
Oce: KINSC H210 (phone: 795.3367)
Course web page: http:/www.haverford.edu/math/cshonkwi/teaching/m318s11
Texts: Understanding Analysis, by Stephen Ab