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, hence E 2 A. On the other hand, if E is in.nite then so is
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] is the diagonal in X Y ,
then
D d d,
D d d, and D d(
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 measure on (X, M).
(a) L1 ( ) = L1 (| |).
(b) If f L1 ( ),
f
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 distinct points with no disjoint
neighborhoods, and consi
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
maximal theorem is essentially sharp.
2. (Problem 24, p
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) F BV ([1, 1]), but G BV ([1, 1]).
/
2. (Problem 35, p
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 inclusion follows from C0 E implies M(C0 ) M(E), it
is sucien
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 problems.
1. Recall 0 < m(E) < and assume 0 < < 1. If no suc
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, . . . be a sequence of disjoint sets in M( ), where M( )
m
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 A M
and f L+ , where IA (x) is the indicator function o
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 the indicator function of A.
1. Since gn (x) fn (x) 0, gn
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: That is, X is a set and
M a -algebra of subsets of X. Dene
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 -algebra. By assumption,
1 (A) = A g1 (x)1 (dx) for A M1 a
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 a constant K such that |x|K |f (y)|dm (1/2) Rn |f (y)|
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 ) = (x2 y 2 )(x2 + y 2 )2 .
(b) f (x, y ) = (1 xy )a (a
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: Show that the latter object is a -algebra.)
2. (Probl
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 measurable functions if we identify functions
that are
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 closed under countable disjoint unions.
We will say tha
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 consult any other
resources besides the the textbook a
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. Consider the sequence xn =
1
.
(2n+1)/2
x2 sin(x2 ) =
n
n
s
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, and work on the remaining
problems by yourself. You will
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 := cfw_B BRn : tB BRn .
n
Show that for every x R and every
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 Lebesgue measure. Show that if = ([0, 1) < , then = m. That
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 . We say that F1 and F2 are the probability density func
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( x y) d y.
Prove that f g is a well-dened function. Prov
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 . Show that if f : X Y is (M1 , N ) measurable, then it is
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
( r n ) 1
n=
1
1
.
, rn +
n
n
so that the complement of
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 computing the integral (i.e. not using the trick from
P
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. (Problem 22, page 59) Let be counting measure on N. Inter