1. A is nonempty because it contains the empty set on X. Obviously if E 2 A then E c 2 A. Furthermore suppose E = [n Ei where E1 ; :; En i=1 are elements of A. Then, if each En is .nite, so is E, henc
Math 5051 - Homework 6
Midterm special (6 questions!)
Due 10/16/08
1. (Problem 46, page 68) Let X = Y = [0, 1], M = N = B[0,1] , = Lebesgue measure,
and = counting measure. If D = cfw_(x, x) : x [0, 1
Math 5051 - Homework 8
Due 10/30/08
1. (Problem 2, page 88) If is a signed measure, E is -null i | |(E ) = 0. Also, if and are
signed measures, i + and .
2. (Problem 3, page 88) Let be a signed measur
Math 5051 - Homework 12 (The last one!)
Due 12/16/08
1. (Problem 32, page 127) A topological space X is Hausdor i every net in X converges to
at most one point. (If X is not Hausdor, let x and y be di
Math 5051 - Homework 10
Due 11/13/08
1. (Problem 22, page 100) If f = 0, there exist C, R > 0 such that Hf (x) C |x|n for
|x| > R. Hence m(cfw_x : Hf (x) > ) C / when is small, so the estimate in the
Math 5051 - Homework 11
Due 11/20/08
1. (Problem 31, page 108) Let F (x) = x2 sin(x1 ) and G(x) = x2 sin(x2 ) for x = 0, and
F (0) = G(0) = 0.
(a) F and G are dierentiable everywhere (including 0).
(b
Ma 5051 Real Variables and Functional Analysis
Solutions for Problem Set #1 due September 10, 2009
Prof. Sawyer Washington University
See m5051hw1.tex for problem text.
1. Since the reverse set inclus
Ma 5051 Real Variables and Functional Analysis
Solutions for Problem Set #3 due September 24, 2009
Prof. Sawyer Washington University
See HOMEWORK#3 on the Math 5051 Web site for the text of the probl
Ma 5051 Real Variables and Functional Analysis
Solutions for Problem Set #2 due September 17, 2009
Prof. Sawyer Washington University
See m5051hw2.tex for problem text.
1. Let cfw_ Aj : j = 1, 2, . .
Ma 5051 Real Variables and Functional Analysis
Solutions for Problem Set #6 due October 15, 2009
Prof. Sawyer Washington University
Let (X, M, ) be a measure space. Recall A f (x)d = IA (x)f (x)d for
Ma 5051 Real Variables and Functional Analysis
Solutions for Problem Set #5 due October 8, 2009
Let (X, M, ) be a measure space. Recall A f (x)d = IA (x)f (x)d for A M
and f L+ L1 , where IA (x) is th
Ma 5051 Real Variables and Functional Analysis
Model Solutions for Problem Set #4 due October 1, 2009
Prof. Sawyer Washington University
In the following, assume that (X, M) is a measurable space: Tha
Ma 5051 Real Variables and Functional Analysis
Solutions for Problem Set #8 due November 12, 2009
Prof. Sawyer Washington University
(With Matt Wallace)
1. Let E M1 M2 , where M1 M2 is the product -al
Ma 5051 Real Variables and Functional Analysis
Solutions for Problem Set #9 due November 19, 2009
Prof. Sawyer Washington University
The measure m(E) below is Lebesgue measure on B(Rn ).
1. (a) Choose
Math 5051 - Homework 7
Due 10/23/08
1. (Problem 55, page 77) Investigate the existence and equality of
11
and 0 0 f (x, y ) dy dx for the following f .
E
f dm2 ,
11
0 0 f (x, y ) dx dy ,
(a) f (x, y )
Math 5051 - Homework 1
Due 9/11/08
1. (Problem 5, page 24) If M is the -algebra generated by E , then M is the union
of the -algebras generated by F as F ranges over all countable subsets of E .
(Hint
Math 5051 - Homework 5
Due 10/09/08
1. (Problem 32, page 63) Suppose (X) < . If f and g are complex-valued
measurable functions on X, dene
|f g|
d.
1 + |f g|
(f, g) =
Then is a metric on the space of
Mathematics 5051, Fall 2013
Solutions to Assignment 1, Problems 4 and 5
Problem 4: Suppose X is some set. We will say that A is of -type if it satises
1. X A
2. A is closed under complements.
3. A is
Mathematics 5051, Fall 2013
Exam 2 (Take-Home), Solutions
Directions: Turn solutions to these problems by 10PM on 13 December 2013. You
may not discuss the problems with anyone besides me. You may not
Mathematics 5051, Fall 2013
Exam 2 (In-Class), Solutions
Problem 1: Explain why the function f dened by f (0) = 0 and f (x) = x2 sin x2 for
x > 0 is not of bounded variation on [0, 1].
Solution. Consi
Mathematics 5051, Fall 2013
Exam 1, 30 October 2013
Directions: You have one hour to do as many problems as you can. After the hour, you
will turn in those problems, take your copy of the exam home, a
Mathematics 5051, Fall 2013
Solutions to Assignment 2, Problems 6,7,8
Problem 6: On Rn , consider the Borel -algebra BRn . For any x Rn , dene
T x := cfw_B BRn : x + B BRn .
For t (0, ), let
D t := cf
Mathematics 5051, Fall 2013
Solutions to Assignment 3, Problems 6,7, and 8
Problem 6: Suppose that is any Borel measure on R which is nite on compact sets and also translation
invariant. Let m is Lebe
Mathematics 5051, Fall 2013
Solutions to Assignment 7, Problems 6 and 7
Problem 6: Let P be a probability measure on a space X . Suppose that f 1 and f 2 are integer-valued measurable functions on X .
Mathematics 5051, Fall 2013
Solutions to Assignment 6, Problems 7 and 8
Problem 7: Suppose that f L1 (R). Let g be continuous with compact support. Dene a new function f g
by
( f g)( x) =
R
f ( y) g(
Mathematics 5051, Fall 2013
Solutions to Assignment 5, Problems 5, 6, 7, 8
Problem 5: Suppose that X is a set and M1 and M2 are -algebras on X such that M1 M2 . Suppose Y is a
set with -algebra N . Sh
Mathematics 5051, Fall 2013
Solutions to Assignment 4, Problems 6,7, and 8
Problem 6: Suppose that ( r n ) 1 is an enumeration of Q and consider the set
n=
rn
I=
n=1
Does there exist an enumeration
(
Mathematics 5051, Fall 2013
Solutions to Assignment 8, Problems 5, 6, and 7
sin x
x [0,) is not integrable. Nonetheless, show that the improper Riemann integral
b sin x
limb 0 x dx converges, without
Math 5051 - Homework 4
Due 10/02/08
1. (Problem 20, page 59) If fn , gn , f, g L1 , fn f and gn g a.e., |fn | gn , and
gn g , then fn f . (Rework the proof of the dominated convergence
theorem.)
2. (P