Math 507/420 Homework Assignment #4
Due in class on Wednesday, November 6
RE-RE-CORRECTED, Oct. 31, 10 PM
SOLUTIONS
1. (a) Show that BR is the smallest -algebra M on R such that every continuous
function f : R R is (M, BR )-measurable.
(b) Given measurabl
Math 421/507: Assignment 2 (Due Friday, Oct.5)
You may use any result from Chapters 0 or 1, or Section 2.1 of Folland, or established
in class.
1. [Borel-Cantelli Lemmas] Let (X, M, ) be a measure space, let cfw_An M be
n=1
a sequence of measurable sets,
Lecture 13:
For f : R R, we will dene f dm as a limit of quantities
of the form:
k
aim(f 1(ai, ai+1])
i=1
(where m is Lebesgue measure). For this to work we need to know
that f 1(ai, ai+1]) L.
Defn: Let (X, M) and (Y, N ) be measurable sets with f : X
Y
Lecture 28:
Fact: (R, L, m) (R, L, m) is not complete.
cfw_0 R is a null set in L L (as above, it is a measurable
rectangle of measure 0).
Let N be the famous non-Lebesgue-measurable set. Then E
cfw_0 N , which is a subset of cfw_0 R, is not L L-meaura
Department of Mathematics
University of British Columbia
MATH 507/420
December 12, 2009, 12 - 2:30
SOLUTIONS
Initials:
Family Name:
I.D. Number:
Question
Signature:
Mark
Out of
1
10
2
15
3
15
4
15
5
15
6
15
7
15
Total
100
CALCULATORS, NOTES OR BOOKS ARE N
Math 507/420 Homework Assignment #1: Due in class on Friday, September 20.
1. Show that a nonempty collection A of subsets is an algebra i 1) for all A, B A,
A B A and 2) for all A A, Ac A.
2. (a) Give an example of the union of two algebras that is not a
Math 507/420 Homework Assignment #3: Due in class on Friday, October 18.
1. Let X be a complete metric space. Show that a subset of X cannot both
contain a countable intersection of dense open sets and
be contained in a countable union of closed nowher
Department of Mathematics
University of British Columbia
MATH 507/420
December 17, 2010, 8:30 - 11:00 AM
Family Name:
Initials:
I.D. Number:
Question
Signature:
Mark
Out of
1
10
2
15
3
15
4
15
5
15
6
15
7
15
Total
100
CALCULATORS, NOTES OR BOOKS ARE NOT P
Math 507/420 Homework Assignment #5
Due in class on Monday, November 25, 2013
1. Let f and f1 , f2 , . . . be measurable functions on a measure space. Show that fn converges in measure to f if and only if for all > 0, there exists N = N such that for
all
Math 507/420 Homework Assignment #2: Due in class on Friday, October 4.
1. Complete the proof of Theorem 1.9, i.e., show that is a complete measure, and is the
unique extension of to M (you may assume, as already proven in class and in the
text, that M is
Math 507/420 Homework Assignment #4
Due in class on Wednesday, November 6
RE-RE-CORRECTED, Oct. 31, 10 PM
1. (a) Show that BR is the smallest -algebra M on R such that every continuous
function f : R R is (M, BR )-measurable.
(b) Given measurable spaces (
Math 507/420 Homework Assignment #6
Due: Monday, December 9 (by 3 PM in the Math Oce (Math Building, Room 121).
1. Let
X = cfw_1, 2, 3, 4.
M = P (X).
be the measure on (X, M) satisfying
(cfw_1) = 1/2, (cfw_2) = 1/3, (cfw_3) = 1/12, (cfw_4) = 1/12.
Let
Real Analysis I - Math 420/507
Fall 2016
Instructor: Malabika Pramanik
Office: Mathematics Building, Room 214
Phone: (604)822-2855
Email: [email protected]
Office hours: To be announced.
Web page: The course website is
http:/www.math.ubc.ca/emala
MAT544 Fall 2009
Homework 10
Problem 1 Using results of HW9 prove that
1. All norms |x| p =
pp Pn
i=1
|xi | p , p 1 and |x| = maxi=1,.,n |xi | are equivalent on
Rn .
2. Define l p as a space of sequences x = (x1 , . . . , xn , . . . ) such that
P
n1
|xn |
Math 507/420 Homework Assignment #6
Due: Monday, December 9 (by 3 PM in the Math Oce (Math Building, Room 121).
1. Let
X = cfw_1, 2, 3, 4.
M = P (X ).
be the measure on (X, M) satisfying
(cfw_1) = 1/2, (cfw_2) = 1/3, (cfw_3) = 1/12, (cfw_4) = 1/12.
L
Math 420/507: Assignment 6 Solutions
Unless otherwise noted, you may use any result from Chapters 0, 1, 2, 3.1-3.3,3.5, or 6.1
of Folland, or established in class.
1. Let be a complex measure on (X, M).
(a) If is real (i.e. a signed measure), show that fo
Math 421/507: Assignment 1 Solutions
1. (a) Show that the intersection of any family of -algebras is a -algebra.
Let A , I be a family of -algebras (indexed by I ), and set A := I A .
If E A, then E A for all , hence E c A for all , hence E c A (closed
un
Math 420/507: Assignment 3 Solutions
1. The point of this problem is to show that the Borel sets on Rn , BRn , are generated
by the family of open rectangles:
R := cfw_(a1 , b1 ) (a2 , b2 ) (an , bn ) | aj , bj R, aj < bj .
A general version of this (whic
Math 420/507: Assignment 5 (Due Friday, Nov. 16)
Unless otherwise noted, you may use any result from Chapters 0, 1, 2 or 3.1-3.3 of
Folland, or established in class.
1. Show that the completions of the measure spaces (R2 , BR BR , m m) and (R2 , L
L, m m
Math 420/507: Assignment 4 Solutions
1. (from Folland 2.3). Fix 0 < a < b. For n = 1, 2, 3, . . ., set fn (x) := aenax benbx .
(a) Show
n=1 0 |fn (x)|dx
= .
Let xn be the unique point in [0, ) where fn (xn ) = 0, that is xn :=
log(b/a)
n(ba) ,
and
set Fn
Review of Measure Theory
Let X be a nonempty set. We denote by P (X ) the set of all subsets of X .
Denition 1 (Algebras)
(a) An algebra is a nonempty collection A of subsets of X such that
i) A, B A = A B A
ii) A A = Ac = X \ A A
(b) A collection A of su
Review of Integration
Denition 1 (Integral) Let (X, M, ) be a measure space and E M.
(a) L+ (X, M) =
f is Mmeasurable . If f L+ (X, M), then
f : X [0, ]
n
ai (Ei E )
f (x) d(x) = sup
n IN, 0 ai < , Ei M for all 1 i n
i=1
E
n
ai Ei (x) f (x) for all x X
an
Review of Signed Measures and the RadonNikodym Theorem
Let X be a nonempty set and M P (X ) a algebra.
Denition 1 (Signed Measures)
(a) A signed measure on (X, M) is a function : M [, ] such that
(i) () = 0
(ii) assumes at most one of the values .
(iii) I
Cardinality
The cardinality of a set provides a precise meaning to the number of elements in the set,
even when the set contains innitely many elements. We start with the following very reasonable
denitions of the set S contains the same number of element
Math 507/420 Homework Assignment #3: Due in class on Friday, October 18.
SOLUTIONS
1. Let X be a complete metric space. Show that a subset of X cannot both
contain a countable intersection of dense open sets and
be contained in a countable union of clo
Math 507/420 Homework Assignment #2: Due in class on Friday, October 4.
SOLUTIONS
1. Complete the proof of Theorem 1.9, i.e., show that is a complete measure, and is the
unique extension of to M (you may assume, as already proven in class and in the
text,
Math 507/420 Homework Assignment #1: Due in class on Friday, September 20.
SOLUTIONS
1. Show that a nonempty collection A of subsets is an algebra i 1) for all A, B A,
A B A and 2) for all A A, Ac A.
Solution:
Only if: follows immediately from the denitio