Regularity of the Lebesgue Stieltjes Measures
Brian Forrest
July 16, 2013
Brian Forrest
Regularity of the Lebesgue Stieltjes Measures
Regularity of the Lebesgue-Stieltjes Measures
Denition: Let A be the algebra of all nite unions of sets of the form
(, b]

Sequences of Functions
Pointwise Convergence
Denition: [Pointwise Convergence]
Let cfw_fn (x) be a sequence of functions dened on an interval
I R. We say that the sequence cfw_fn (x) converges pointwise on I
to a function f0 (x) : I R, if for each x0 I ,

Integrability of Continuous Functions
Problem: If ﬂat) is continuous on [a, b], is ﬂat) also integrable on
[at b]?
Recall: Theorem: Assume that ﬂat) is bounded on [aE b]. Then
ﬂat) is integrable on [a, b] if and only if for every E 1: 0, there
exists a

Two Criteria for Integrability
Denition of the Integral
Denition: [Riemann Integral]
Assume that f (x) is bounded on [a, b]. We say that f (x) is
Riemann integrable on [a, b] if
b
b
f (x) dx =
f (x) dx,
a
a
in which case we denote this common value by
b
f

Formal Denition of the Integral
Formal Denition of the Integral
Brian Forrest
April 7, 2011
Brian Forrest
Formal Denition of the Integral
Formal Denition of the Integral
Upper and Lower Riemann Sums
Denition: [Upper and Lower Riemann Sums]
Assume that f (

Upper and Lower Riemann Sums
Upper and Lower Riemann Sums
Problem: Given two dierent partitions P1 and P2 is
Lb (f , P1 ) Ub (f , P2 )?
a
a
Brian Forrest
Upper and Lower Riemann Sums
Upper and Lower Riemann Sums
Renement Theorem
a
P Q
b
Denition: [Renemen

Upper and Lower Riemann Sums
Partitions
Denition: [Partition] Let [a, b] be a nite, closed interval. A
partition P of [a, b] is a nite subset of [a, b] of the form
P = cfw_ti : a = t0 < t1 < < tn = b.
For a partition P of [a, b] having n elements,
1. dene

1
PMATH 451 : Measure and Integration:
Fall 2014
Electronic Assignment #6
Due by 11:00 pm EST on Sunday, November 23 , 2014
Instructions:
Prior to attempting this assignment, you should have reviewed the course notes for Chapter 7.
You should also have v

1
PMATH 451 : Measure and Integration:
Fall 2014
Electronic Assignment #5
Due by 11:00 pm EST on Sunday, November 16 , 2014
Instructions:
Prior to attempting this assignment, you should have reviewed the course notes for Chapter 5.
You should also have v

1
PMATH 451 : Measure and Integration:
Fall 2014
Electronic Assignment #4
Due by 11:00 pm EST on Sunday, October 26, 2014
Instructions:
Prior to attempting this assignment, you should have reviewed the course notes for Chapter 4 and
the rst three section

1
PMATH 451 : Measure and Integration:
Fall 2014
Electronic Assignment #3
Due by 11:00 pm EST on Sunday, October 19, 2014
Instructions:
Prior to attempting this assignment, you should have reviewed the course notes for Chapter 3,
and also viewed the lect

1
PMATH 451 : Measure and Integration:
Fall 2014
Electronic Assignment #2
Due by 11:00 pm EST on Sunday, September 28, 2014
Instructions:
Prior to attempting this assignment, you should have reviewed the course notes for Chapter 2, and
also viewed the le

Sequences of Functions
Flaws in Pointwise Convergence
fn (x) = x n f0 (x) =
Brian Forrest
Sequences of Functions
Sequences of Functions
Flaws in Pointwise Convergence
fn (x) = x n f0 (x) =
Brian Forrest
Sequences of Functions
Sequences of Functions
Flaws

Vector Valued Integrals
Brian Forrest
May 28, 2013
Brian Forrest
Vector Valued Integrals
Denition
Goal:
Let B be a Banach space and that F : [a, b] B is a continuous
function. In this section we will see how we may dene a
vector-valued version of the Riem

Algebras and -algebras
Brian Forrest
July 16, 2013
Brian Forrest
Algebras and -algebras
Algebras and -algebras
Recall: In summarizing the aws of the Riemann integral we can
focus on two main points:
1) Many nice functions are not Riemann integrable.
2) Th

Lebesgue Stieltjes measures
Brian Forrest
July 16, 2013
Brian Forrest
Lebesgue Stieltjes measures
Measures on B(R)
Recall: The Carathodory Extension Theorem allowed us to extend
e
measure from an algebra A to the -algebra generated by A.
Remark: Let A be

Extension Theorems
Brian Forrest
July 3, 2013
Brian Forrest
Extension Theorems
Carathodory Extension Theorem
e
Denition: The measure constructed in the previous theorem is
called the Carathodory extension of the measure .
e
Problem: Given a measure on an

Extension Theorems
Brian Forrest
July 3, 2013
Brian Forrest
Extension Theorems
Measures on Algebras
Recall: The Carathodory Method allowed us to construct a
e
measure from an outer measure.
The Lebesgue outer measure m was dened on P(R) by E R by
(In ) |

Lebesgue Measure: Part II
Brian Forrest
July 16, 2013
Brian Forrest
Lebesgue Measure: Part II
Lebesgue Measure
Recall: The Lebesgue outer measure m was dened on P(R) by
E R by
m (E ) = inf
(In ) | E
n=1
In , In s are open intervals
n=1
for any E R.
Denit

Lebesgue Measure
Brian Forrest
July 16, 2013
Brian Forrest
Lebesgue Measure
Lebesgue Measure
Recall: The Lebesgue outer measure m was dened on P(R) by
E R by
m (E ) = inf
(In ) | E
n=1
In , In s are open intervals
n=1
for any E R.
Denition: We denote the

Proof of the Carathodory Theorem
e
Brian Forrest
July 16, 2013
Brian Forrest
Proof of the Carathodory Theorem
e
Outer Measures and Measurable Sets
Denition: [Outer Measure]
Let X be any non-empty set. A function : P(X ) R is called an
outer measure if
1)

Outer Measures and the Carathodory Method
e
Brian Forrest
July 16, 2013
Brian Forrest
Outer Measures and the Carathodory Method
e
Goal
Recall: In looking for a generalized notion of length for subsets of
R we ask for a measure m on P(R) with the following

Basic Properties of Measures
Brian Forrest
July 16, 2013
Brian Forrest
Basic Properties of Measures
Monotonicity
Proposition: [Monotonicity]
Let (X , A, ) be a measure space. If E , F A with E F , then
(E ) (F ). If (E ) < , then (F \ E ) = (F ) (E ).
Pro

Basic Properties of Measures
Brian Forrest
June 7, 2013
Brian Forrest
Basic Properties of Measures
Basic Properities of Measures
Denition: A pair (X , A) consisting set X together with a -algebra
A P(X ) is called a measurable space.
A measure on A is a f

Introduction to Measures
Brian Forrest
June 26, 2013
Brian Forrest
Introduction to Measures
Two problems:
Problem 1: Is there a way to extend the notions of length, area or
volume to arbitrary subsets of Rn .
Problem 2: Can the notions of length, area or

Borel Sets in R and Baires Category Theorem
Brian Forrest
June 26, 2013
Brian Forrest
Borel Sets in R and Baires Category Theorem
Two problems:
Problem 1: Are there sets in R which are neither F or G ?
Fact: |B(R)| = c, the cardinality of R most sets are

1
PMATH 451 : Measure and Integration:
Fall 2014
Electronic Assignment #1
Due by 11:00 pm EST on Sunday, September 21, 2014
Instructions:
Prior to attempting this assignment, you should have reviewed the course notes for Chapter 1 and
Chapter 2, and also

1
PMATH 451 : Measure and Integration:
Fall 2014
Written Assignment #3
Due by 11:00 pm EDT on Sunday, November 30, 2014
Instructions:
Read and complete as much of the following assignment as possible.
ONLY THOSE PROBLEMS OR PARTS THERE OF MARKED WITH AN

PMATH 451 Final Examination Information
1) The exam covers material up to and including Fubinis Theorem. The
first Chapter on the Riemann Integral is not included though you are
responsible for all assignment problems relating to the Riemann integral.
2)

Chapter 5
The Space M eas(X, A)
In this chapter, we will study the properties of the space of finite signed measures on a measurable space
(X, A). In particular we will show that with respect to a very natural norm that this space is in fact a
Banach spac

Chapter 4
Signed Measures
Up until now our measures have always assumed values that were greater than or equal to 0. In this chapter
we will extend our definition to allow for both positive and negative values.
4.1
Basic Properties of Signed Measures
Defi

1
PMATH 451 : Measure and Integration:
Fall 2013
Electronic Assignment #5
Due by 11:00 pm EST on Sunday, November 16 , 2013
Instructions:
Prior to attempting this assignment, you should have reviewed the course notes for Chapter 5.
You should also have v

1
PMATH 451 : Measure and Integration
Fall 2013
Electronic Assignment #4 Solutions
1) b) False In general P N would simply be a null set.
2) a) True (The Hahn Decomposition need not be uniique, but the Jordan Decomposition is.)
3) a) True
4) a) True
5) a)

1
I think we are all familiar with the sequence space
`1 (N) = cfw_an |
X
|an | < .
n=1
Now since every sequence is really a function f : N R given by
f (n) = an
we can really re-write our space above as:
`1 (N) = cfw_f : N R |
X
|f (n)| < .
n=1
The prob

Chapter 6
Riesz Representation Theorems
6.1
Dual Spaces
Definition 6.1.1. Let V and W be vector spaces over R. We let
L(V, W ) = cfw_T : V W | T is linear.
The space L(V, R) is denoted by V ] and elements of V ] are called linear functionals.
Example 6.1.