MAT 324: Real Analysis, Fall 2017
Solutions to Problem Set 10
Problem 1 (16pts)
Suppose X is a set and : 2X [0, ] is a function such that
() = 0,
(A) (B) if A B,
and
[
An
n=1
X
(An ) A1 , A2 , . .
MAT 324: Real Analysis, Fall 2017
Homework Assignment 6
Please read carefully Sections 4.4-4.6 in the textbook, prove all propositions and do all exercises
you encounter along the way, and write up cl
MAT 324: Real Analysis, Fall 2017
Midterm (80 mins)
Please write your solutions on the provided printer paper, starting each problem on a new page
and clearly indicating the number (and part) of the p
MAT 324: Real Analysis, Fall 2017
Solutions to Problem Set 8
Problem 1 (10pts)
Let (V, h, i) be an inner-product space over C and e1 , e2 , . . . V be a sequence of orthonormal
vectors, i.e.
(
1, if i
MAT 324: Real Analysis, Fall 2017
Homework Assignment 7
Please read carefully Sections 5.1 and 5.3 in the textbook, prove all propositions and do all exercises
you encounter along the way, and write u
MAT 324: Real Analysis, Fall 2017
Basic Review Questions I
1. Example of monotone class on Z+ , which is not a -field:
2. Example of field on Z+ , which is not a -field:
3. Example of -field on Z+ :
(
MAT 324: Real Analysis, Fall 2017
Homework Assignment 2
This homework assignment covers 1.5 weeks.
Please read carefully Sections 2.3-2.7 and page 302 in the textbook, prove all propositions and do
al
MAT 324: Real Analysis, Fall 2017
Homework Assignment 1
Please read carefully Chapter 1 and Sections 2.1 and 2.2 of the textbook, prove all propositions and
do all exercises you encounter along the wa
MAT 324: Real Analysis, Fall 2017
Solutions to Problem Set 2
Problem 1 (5pts)
Suppose A B C R are such that A, C M and m(A) = m(C) < . Show that B M and
m(A) = m(B) = m(C).
Since A, C M, A C, and m(A)
MAT 324: Real Analysis, Fall 2017
Solutions to Problem Set 5
Problem 1 (5pts)
Let (X, F, ) be a measure space and fn : X [0, ] be a sequence of measurable functions.
Show that
Z X
Z
X
fn d .
fn d =
X
MAT 324: Real Analysis, Fall 2017
Course Information
Teaching Staff
Instructor: Aleksey Zinger ([email protected]); OHs: Tu 2:30-4pm, Th 11:30am-1pm in Math 3-111
TA: Yuhan Sun ([email protected]);
OHs: W 4-5p
MAT 324: Real Analysis, Fall 2017
Homework Assignment 10
This homework assignment covers 1.5 weeks.
Please read carefully Sections 6.3-6.6 in the textbook and pp164-170 in Rudins book (see website),
p
MAT 324: Real Analysis, Fall 2017
Solutions to Problem Set 3
Problem 1 (25pts)
Let (X, F, ) be a measure space and F be the completion of F with respect to as defined in
Section 2.5. Show that
(a) F
MAT 324: Real Analysis, Fall 2017
Solutions to Problem Set 11
Problem 1 (8pts)
Let , 1 , 2 be measures on a measurable space (X, F). Show that
(a) if 1 , 2 < , then 1 +2 < .
(b) if 1 , 2 , then 1 +2 .
MAT 324: Real Analysis, Fall 2017
Final Exam (150 mins)
Name:
There are 9 problems on this exam of varying difficulty: Problem 1 below, Problem 2 on the back
of this cover sheet, and Problems 3-9 on a
MAT 324: Real Analysis, Fall 2017
Solutions to Problem Set 9
Problem 1 (8pts)
Let (X, F, ) be a measure space. Suppose there exist A, B F such that AB = and (A), (B) R+ .
Show that the norm kkp on Lp
MAT 324: Real Analysis, Fall 2017
Solutions to Problem Set 1
Problem 1 (5pts)
Is the function
X
f (x) =
2n sin(2n x)
n=0
Riemann integrable on [0, 2]? Justify your answer.
Yes, because the finite sum
MAT 324: Real Analysis, Fall 2017
Homework Assignment 11
Please read carefully Sections 7.1 and 7.2 in the textbook, prove all propositions and do all exercises
you encounter along the way, and write
MAT 324: Real Analysis, Fall 2017
Homework Assignment 4
Please read carefully Sections 3.4-3.6,4.1 and page 303 in the textbook, prove all propositions and
do all exercises you encounter along the way
MAT 324: Real Analysis, Fall 2017
Final Exam Information
The final exam will take place on Friday, December 15, 11:15am-1:45pm in ESS 181. It will cover
Chapters 1-6 (except for Sections 2.6, 3.5, 4.7
MAT 324: Real Analysis, Fall 2017
Homework Assignment 5
Please read carefully Sections 4.2 and 4.3 in the textbook, prove all propositions and do all exercises
you encounter along the way, and write u
MAT 324: Real Analysis, Fall 2017
Solutions to Problem Set 7
Problem 1 (8pts)
Let (X, F, ) be a measure space and p, q, r [1, ] be such that 1/p+1/q = 1/r. Show that
f g kf kp kgkq
r
for all measura
MAT 324: Real Analysis, Fall 2017
Bonus Homework Assignment
Please read carefully Sections 4.6-4.8 in the textbook. Well start on Chapter 5 on Tuesday, 10/24.
Bonus Problem Set (due in class on Tuesda
MAT 324: Real Analysis, Fall 2017
Homework Assignment 9
Please read carefully Sections 5.2 and 6.1-6.3 in the textbook and pp160-164 in Rudins book (see
website), prove all propositions and do all exe
MAT 324: Real Analysis, Fall 2017
Comments on and Additional Corrections to the Textbook
November 29, 2017
The corrections listed below are in addition to those listed in the official erratum provided
MAT 324: Real Analysis, Fall 2017
Solutions to Problem Set 4
Problem 1 (10pts)
X
an (x)
with each an (x) cfw_0, 1 taking the infinite expansion for
2n
all x 6= 0. Show that the function an : [0, 1] R
MAT 324: Real Analysis, Fall 2017
Homework Assignment 3
Please read carefully Sections 3.1-3.4 in the textbook, prove all propositions and do all exercises
you encounter along the way, and write up cl
MAT 324: Real Analysis, Fall 2017
Solutions to Midterm
Problem 1 (10pts)
(a) Give a definition of -field on a set X.
(b) Describe all -fields on the set X cfw_a, b of two elements; explain why there a
MAT 324: Real Analysis, Fall 2017
Solutions to Problem Set 6
Problem 1 (8pts)
Let (X, F, ) be a measure space and fn : X R be a sequence of measurable functions converging
almost everywhere to a funct
MAT324: Real Analysis Fall 2014
Assignment 6
Due Thursday, November 6, in class.
Problem 1: Which of the following statements are true and which are false? Explain.
a) L1 (R) L2 (R)
b) L2 (R) L1 (R)
c
Axiom of Countable Choice
For each n Z+ , let An be a nonempty set. ACC says that there exists a sequence (an )nZ+ with
an An for each n Z+ . If Z+ were replaced by a finite set, this would be easily
MAT 324: Real Analysis, Fall 2017
Solutions to Bonus Problem Set
Problem 1 (10pts)
Complete the following definitions.
(a) An equivalence relation on a set X is a subset A X 2 such that
(i) (x, x) A x
MAT 324: Real Analysis, Fall 2017
First-Day Quiz (40 mins)
Name:
1 (1pt) 10ln 6 6ln 10
(A) 0
Answer only
equals
(B)
1
(C)
1
(D)
4 ln 4
1
256
(E)
2 (1pt) The graph of the function y = (x2)3 +3 is obtai
MAT 324: Real Analysis, Fall 2017
Homework Assignment 8
Please read carefully Sections 8.1, 5.2, 5.4, and 5.5 in the textbook, prove all propositions and do
all exercises you encounter along the way,
1
PAGE 164
If A = A1 A2 , then the function 2 7 P (A2 ) is a step function:
P (A1 ) if 2 A2
P (A2 ) =
0
if 2
/ A2
and hence we have
P (A) = P1 (A1 )P2 (A2 ) =
Z
P (A2 ) dP2 (2 ).
2
replaced by
If A =