measuretheory demystified
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Primbs, MS&E345 1
Measure Theory in a Lecture
Monday, March 19, 2012
Primbs, MS&E345 2
Perspective
σ-Algebras
Measurable Functions
Measure and Integration
Radon-Nikodym Theorem
Riesz Representation Theorem
Measure Theory
Probability Theory
Monday, March 19, 2012
Primbs, MS&E345 2
Perspective
σ-Algebras
Measurable Functions
Measure and Integration
Radon-Nikodym Theorem
Riesz Representation Theorem
Measure Theory
Probability Theory
Monday, March 19, 2012
Primbs, MS&E345 3
A little perspective:
Monday, March 19, 2012
Primbs, MS&E345 3
A little perspective:
Riemann did this...
Monday, March 19, 2012
Primbs, MS&E345 3
A little perspective:
Riemann did this...
...
Monday, March 19, 2012
Primbs, MS&E345 4
A little perspective:
Monday, March 19, 2012
Primbs, MS&E345 4
A little perspective:
Lebesgue did this...
Monday, March 19, 2012
Primbs, MS&E345 4
A little perspective:
Lebesgue did this...
Monday, March 19, 2012
Primbs, MS&E345 4
A little perspective:
Lebesgue did this...
Monday, March 19, 2012
Primbs, MS&E345 4
A little perspective:
Lebesgue did this...
Monday, March 19, 2012
Primbs, MS&E345 4
A little perspective:
Lebesgue did this...
Monday, March 19, 2012
Primbs, MS&E345 4
A little perspective:
Lebesgue did this...
Monday, March 19, 2012
Primbs, MS&E345 4
A little perspective:
Lebesgue did this...
Monday, March 19, 2012
Primbs, MS&E345 5
A little perspective:
Lebesgue did this...
Why is this better????
Monday, March 19, 2012
Primbs, MS&E345 6
A little perspective:
Lebesgue did this...
Why is this better????
Monday, March 19, 2012
Primbs, MS&E345 6
A little perspective:
Lebesgue did this...
Why is this better???? Consider the function:
Monday, March 19, 2012
Primbs, MS&E345 6
A little perspective:
Lebesgue did this...
Why is this better???? Consider the function:
Monday, March 19, 2012
Primbs, MS&E345 6
A little perspective:
Lebesgue did this...
Why is this better???? Consider the function:
If we follow Lebesgue’s reasoning, then the integral of this function over the set [0,1] should be:
(Height) x (width of irrationals) = 1 x (measure of irrationals)
Height
Width of Irrationals
Monday, March 19, 2012
Primbs, MS&E345 6
A little perspective:
Lebesgue did this...
Why is this better???? Consider the function:
If we follow Lebesgue’s reasoning, then the integral of this function over the set [0,1] should be:
(Height) x (width of irrationals) = 1 x (measure of irrationals)
Height
Width of Irrationals
If we can “measure” the size of superlevel sets, we can integrate a lot of function!
Monday, March 19, 2012
Primbs, MS&E345 7
Monday, March 19, 2012
Primbs, MS&E345 7
Measure Theory begins from this simple motivation.
Monday, March 19, 2012
Primbs, MS&E345 7
Measure Theory begins from this simple motivation.
If you remember a couple of simple principles, measure and integration theory becomes quite intuitive.
Monday, March 19, 2012
Primbs, MS&E345 7
Measure Theory begins from this simple motivation.
If you remember a couple of simple principles, measure and integration theory becomes quite intuitive.
The first principle is this...
Monday, March 19, 2012
Primbs, MS&E345 7
Measure Theory begins from this simple motivation.
If you remember a couple of simple principles, measure and integration theory becomes quite intuitive.
The first principle is this...
Integration is about functions. Measure theory is about sets. The connection between functions and sets is super/sub-level sets.
Monday, March 19, 2012
Primbs, MS&E345 8
Perspective
σ-Algebras
Measurable Functions
Measure and Integration
Radon-Nikodym Theorem
Riesz Representation Theorem
Measure Theory
Probability Theory
Monday, March 19, 2012
Primbs, MS&E345 8
Perspective
σ-Algebras
Measurable Functions
Measure and Integration
Radon-Nikodym Theorem
Riesz Representation Theorem
Measure Theory
Probability Theory
Monday, March 19, 2012
Primbs, MS&E345 9
Monday, March 19, 2012
Primbs, MS&E345 9
Measure theory is about measuring the size of things.
Monday, March 19, 2012
Primbs, MS&E345 9
Measure theory is about measuring the size of things.
In math we measure the size of sets.
Set A
Monday, March 19, 2012
Primbs, MS&E345 9
Measure theory is about measuring the size of things.
In math we measure the size of sets.
Set AHow big is the set A?
Monday, March 19, 2012
Primbs, MS&E345 10
Monday, March 19, 2012
Primbs, MS&E345 10
“Size” should have the following property:
Monday, March 19, 2012
Primbs, MS&E345 10
“Size” should have the following property:
The size of the union of disjoint sets should equal the sum of the sizes of the individual sets.
Monday, March 19, 2012
Primbs, MS&E345 10
“Size” should have the following property:
The size of the union of disjoint sets should equal the sum of the sizes of the individual sets.
Makes sense.... but there is a problem with this...
Monday, March 19, 2012
Primbs, MS&E345 10
“Size” should have the following property:
The size of the union of disjoint sets should equal the sum of the sizes of the individual sets.
Makes sense.... but there is a problem with this...
Consider the interval [0,1]. It is the disjoint union of all real numbers between 0 and 1. Therefore, according to above, the size of [0,1] should be the sum of the sizes of a single real number.
Monday, March 19, 2012
Primbs, MS&E345 10
“Size” should have the following property:
The size of the union of disjoint sets should equal the sum of the sizes of the individual sets.
Makes sense.... but there is a problem with this...
Consider the interval [0,1]. It is the disjoint union of all real numbers between 0 and 1. Therefore, according to above, the size of [0,1] should be the sum of the sizes of a single real number.
If the size of a singleton is 0 then the size of [0,1] is 0.
Monday, March 19, 2012
Primbs, MS&E345 10
“Size” should have the following property:
The size of the union of disjoint sets should equal the sum of the sizes of the individual sets.
Makes sense.... but there is a problem with this...
Consider the interval [0,1]. It is the disjoint union of all real numbers between 0 and 1. Therefore, according to above, the size of [0,1] should be the sum of the sizes of a single real number.
If the size of a singleton is 0 then the size of [0,1] is 0.If the size of a singleton is non-zero, then the size of [0,1] is infinity!
Monday, March 19, 2012
Primbs, MS&E345 10
“Size” should have the following property:
The size of the union of disjoint sets should equal the sum of the sizes of the individual sets.
Makes sense.... but there is a problem with this...
Consider the interval [0,1]. It is the disjoint union of all real numbers between 0 and 1. Therefore, according to above, the size of [0,1] should be the sum of the sizes of a single real number.
If the size of a singleton is 0 then the size of [0,1] is 0.If the size of a singleton is non-zero, then the size of [0,1] is infinity!
This shows we have to be a bit more careful.
Monday, March 19, 2012
Primbs, MS&E345 11
Sigma Algebras (σ-algebra)
Monday, March 19, 2012
Primbs, MS&E345 11
Sigma Algebras (σ-algebra)Consider a set Ω.
Monday, March 19, 2012
Primbs, MS&E345 11
Sigma Algebras (σ-algebra)Consider a set Ω.
Let F be a collection of subsets of Ω.
Monday, March 19, 2012
Primbs, MS&E345 11
Sigma Algebras (σ-algebra)Consider a set Ω.
Let F be a collection of subsets of Ω.
F is a σ-algebra if:
Monday, March 19, 2012
Primbs, MS&E345 11
Sigma Algebras (σ-algebra)Consider a set Ω.
Let F be a collection of subsets of Ω.
F is a σ-algebra if:
1)
Monday, March 19, 2012
Primbs, MS&E345 11
Sigma Algebras (σ-algebra)Consider a set Ω.
Let F be a collection of subsets of Ω.
F is a σ-algebra if:
1)
2)
Monday, March 19, 2012
Primbs, MS&E345 11
Sigma Algebras (σ-algebra)Consider a set Ω.
Let F be a collection of subsets of Ω.
F is a σ-algebra if:
1)
2)
3)
Monday, March 19, 2012
Primbs, MS&E345 11
Sigma Algebras (σ-algebra)Consider a set Ω.
Let F be a collection of subsets of Ω.
F is a σ-algebra if:
1)
2)
3)
4)
Monday, March 19, 2012
Primbs, MS&E345 11
Sigma Algebras (σ-algebra)Consider a set Ω.
Let F be a collection of subsets of Ω.
F is a σ-algebra if: An algebra of sets is closed under finite set operations. A σ-algebra is closed under countable set operations. In mathematics, σ often refers to “countable”.
1)
2)
3)
4)
Monday, March 19, 2012
Primbs, MS&E345 12
Perspective
σ-Algebras
Measurable Functions
Measure and Integration
Radon-Nikodym Theorem
Riesz Representation Theorem
Measure Theory
Probability Theory
Monday, March 19, 2012
Primbs, MS&E345 12
Perspective
σ-Algebras
Measurable Functions
Measure and Integration
Radon-Nikodym Theorem
Riesz Representation Theorem
Measure Theory
Probability Theory
Monday, March 19, 2012
Primbs, MS&E345 13
Algebras are what we need for super/sub-level sets of a vector space of indicator functions.
Monday, March 19, 2012
Primbs, MS&E345 13
Algebras are what we need for super/sub-level sets of a vector space of indicator functions.
Monday, March 19, 2012
Primbs, MS&E345 13
Algebras are what we need for super/sub-level sets of a vector space of indicator functions.
Monday, March 19, 2012
Primbs, MS&E345 13
Algebras are what we need for super/sub-level sets of a vector space of indicator functions.
Monday, March 19, 2012
Primbs, MS&E345 13
Algebras are what we need for super/sub-level sets of a vector space of indicator functions.
Superlevel Sets
Monday, March 19, 2012
Primbs, MS&E345 13
Algebras are what we need for super/sub-level sets of a vector space of indicator functions.
Superlevel Sets
Monday, March 19, 2012
Primbs, MS&E345 13
Algebras are what we need for super/sub-level sets of a vector space of indicator functions.
Superlevel Sets
Monday, March 19, 2012
Primbs, MS&E345 14
Superlevel Sets
Monday, March 19, 2012
Primbs, MS&E345 14
Superlevel Sets
Simple Functions
Monday, March 19, 2012
Primbs, MS&E345 14
Superlevel Sets
Simple FunctionsSimple functions are finite linear combinations of of indicator functions.
Monday, March 19, 2012
Primbs, MS&E345 14
Superlevel Sets
Simple FunctionsSimple functions are finite linear combinations of of indicator functions.
Monday, March 19, 2012
Primbs, MS&E345 14
Superlevel Sets
σ−algebras let us take limits of these. That is more interesting!!
Simple FunctionsSimple functions are finite linear combinations of of indicator functions.
Monday, March 19, 2012
Primbs, MS&E345 15
Monday, March 19, 2012
Primbs, MS&E345 15
Measurable Functions
Monday, March 19, 2012
Primbs, MS&E345 15
Measurable Functions
Given a Measurable Space (Ω,F)
Monday, March 19, 2012
Primbs, MS&E345 15
Measurable Functions
Given a Measurable Space (Ω,F)
We say that a function: is measurable with respect to F if:
Monday, March 19, 2012
Primbs, MS&E345 15
Measurable Functions
Given a Measurable Space (Ω,F)
We say that a function: is measurable with respect to F if:
Monday, March 19, 2012
Primbs, MS&E345 15
Measurable Functions
Given a Measurable Space (Ω,F)
We say that a function: is measurable with respect to F if:
This simply says that a function is measurable with respect to the σ-algebra if all its superlevel sets are in the σ-algebra.
Monday, March 19, 2012
Primbs, MS&E345 15
Measurable Functions
Given a Measurable Space (Ω,F)
We say that a function: is measurable with respect to F if:
This simply says that a function is measurable with respect to the σ-algebra if all its superlevel sets are in the σ-algebra.
Hence, a measurable function is one that when we slice it like Lebesgue, we can measure the “width” of the part of the function above the slice. Simple...right?
Monday, March 19, 2012
Primbs, MS&E345 16
Monday, March 19, 2012
Primbs, MS&E345 16
Another equivalent way to think of measurable functions is as the pointwise limit of simple functions.
Monday, March 19, 2012
Primbs, MS&E345 16
Another equivalent way to think of measurable functions is as the pointwise limit of simple functions.
In fact, if a measurable function is non-negative, we can say it is the increasing pointwise limit of simple functions.
Monday, March 19, 2012
Primbs, MS&E345 16
Another equivalent way to think of measurable functions is as the pointwise limit of simple functions.
In fact, if a measurable function is non-negative, we can say it is the increasing pointwise limit of simple functions.
This is simply going back to Lebesgue’s picture...
Monday, March 19, 2012
Primbs, MS&E345 17
Intuition behind measurable functions.
Monday, March 19, 2012
Primbs, MS&E345 17
Intuition behind measurable functions.
They are “constant on the sets in the sigma algebra.”
Monday, March 19, 2012
Primbs, MS&E345 17
Intuition behind measurable functions.
They are “constant on the sets in the sigma algebra.”
Monday, March 19, 2012
Primbs, MS&E345 17
Intuition behind measurable functions.
They are “constant on the sets in the sigma algebra.”
Monday, March 19, 2012
Primbs, MS&E345 17
Intuition behind measurable functions.
They are “constant on the sets in the sigma algebra.”
What do measurable functions look like?
Monday, March 19, 2012
Primbs, MS&E345 17
Intuition behind measurable functions.
They are “constant on the sets in the sigma algebra.”
What do measurable functions look like?
Monday, March 19, 2012
Primbs, MS&E345 17
Intuition behind measurable functions.
They are “constant on the sets in the sigma algebra.”
What do measurable functions look like?Yes
Monday, March 19, 2012
Primbs, MS&E345 17
Intuition behind measurable functions.
They are “constant on the sets in the sigma algebra.”
What do measurable functions look like?Yes
Monday, March 19, 2012
Primbs, MS&E345 17
Intuition behind measurable functions.
They are “constant on the sets in the sigma algebra.”
What do measurable functions look like?Yes Yes
Monday, March 19, 2012
Primbs, MS&E345 17
Intuition behind measurable functions.
They are “constant on the sets in the sigma algebra.”
What do measurable functions look like?Yes Yes
Monday, March 19, 2012
Primbs, MS&E345 17
Intuition behind measurable functions.
They are “constant on the sets in the sigma algebra.”
What do measurable functions look like?Yes Yes No
Monday, March 19, 2012
Primbs, MS&E345 17
Intuition behind measurable functions.
They are “constant on the sets in the sigma algebra.”
What do measurable functions look like?Yes Yes No
Intuitively speaking, Measurable functions are constant on the sets in the σ-alg.
Monday, March 19, 2012
Primbs, MS&E345 17
Intuition behind measurable functions.
They are “constant on the sets in the sigma algebra.”
What do measurable functions look like?Yes Yes No
Intuitively speaking, Measurable functions are constant on the sets in the σ-alg.
More accurately, they are limits of functions are constant on the sets in the σ-algebra. The σ in σ-algebra gives us this.
Monday, March 19, 2012
Primbs, MS&E345 18
“Information” and σ-Algebras.
Monday, March 19, 2012
Primbs, MS&E345 18
“Information” and σ-Algebras.
Since measurable functions are “constant” on the σ-algebra, if I am trying to determine information from a measurable function, the σ-algebra determines the information that I can obtain.
Monday, March 19, 2012
Primbs, MS&E345 18
“Information” and σ-Algebras.
Since measurable functions are “constant” on the σ-algebra, if I am trying to determine information from a measurable function, the σ-algebra determines the information that I can obtain. I measure What is the most information that I
determine about ω?
Monday, March 19, 2012
Primbs, MS&E345 18
“Information” and σ-Algebras.
Since measurable functions are “constant” on the σ-algebra, if I am trying to determine information from a measurable function, the σ-algebra determines the information that I can obtain. I measure What is the most information that I
determine about ω? The best possible I can do is to say that with
Monday, March 19, 2012
Primbs, MS&E345 18
“Information” and σ-Algebras.
Since measurable functions are “constant” on the σ-algebra, if I am trying to determine information from a measurable function, the σ-algebra determines the information that I can obtain. I measure What is the most information that I
determine about ω?
σ-algebras determine the amount of information possible in a function.
The best possible I can do is to say that with
Monday, March 19, 2012
Primbs, MS&E345 19
Perspective
σ-Algebras
Measurable Functions
Measure and Integration
Radon-Nikodym Theorem
Riesz Representation Theorem
Measure Theory
Probability Theory
Monday, March 19, 2012
Primbs, MS&E345 19
Perspective
σ-Algebras
Measurable Functions
Measure and Integration
Radon-Nikodym Theorem
Riesz Representation Theorem
Measure Theory
Probability Theory
Monday, March 19, 2012
Primbs, MS&E345 20
Measures and Integration
Monday, March 19, 2012
Primbs, MS&E345 20
Measures and Integration
A measurable space (Ω,F) defines the sets can be measured.
Monday, March 19, 2012
Primbs, MS&E345 20
Measures and Integration
A measurable space (Ω,F) defines the sets can be measured.
Now we actually have to measure them...
Monday, March 19, 2012
Primbs, MS&E345 20
Measures and Integration
A measurable space (Ω,F) defines the sets can be measured.
Now we actually have to measure them...
What are the properties that “size” should satisfy.
Monday, March 19, 2012
Primbs, MS&E345 20
Measures and Integration
A measurable space (Ω,F) defines the sets can be measured.
Now we actually have to measure them...
What are the properties that “size” should satisfy.
If you think about it long enough, there are really only two...
Monday, March 19, 2012
Primbs, MS&E345 21
Definition of a measure:
Monday, March 19, 2012
Primbs, MS&E345 21
Definition of a measure:Given a Measurable Space (Ω,F),
Monday, March 19, 2012
Primbs, MS&E345 21
Definition of a measure:Given a Measurable Space (Ω,F),
A measure is a functionsatisfying two properties:
Monday, March 19, 2012
Primbs, MS&E345 21
Definition of a measure:Given a Measurable Space (Ω,F),
A measure is a functionsatisfying two properties:
1)
Monday, March 19, 2012
Primbs, MS&E345 21
Definition of a measure:Given a Measurable Space (Ω,F),
A measure is a functionsatisfying two properties:
1)
2)
Monday, March 19, 2012
Primbs, MS&E345 21
Definition of a measure:Given a Measurable Space (Ω,F),
A measure is a functionsatisfying two properties:
1)
2)
(2) is known as “countable additivity”. However, I think of it as “linearity and left continuity for sets”.
Monday, March 19, 2012
Primbs, MS&E345 22
Monday, March 19, 2012
Primbs, MS&E345 22
Fundamental properties of measures (or “size”):
Monday, March 19, 2012
Primbs, MS&E345 22
Fundamental properties of measures (or “size”):
Left Continuity: This is a trivial consequence of the definition!
Monday, March 19, 2012
Primbs, MS&E345 22
Fundamental properties of measures (or “size”):
Left Continuity: This is a trivial consequence of the definition!
Let then
Monday, March 19, 2012
Primbs, MS&E345 22
Fundamental properties of measures (or “size”):
Left Continuity: This is a trivial consequence of the definition!
Let then
Monday, March 19, 2012
Primbs, MS&E345 22
Fundamental properties of measures (or “size”):
Left Continuity: This is a trivial consequence of the definition!
Let then
Monday, March 19, 2012
Primbs, MS&E345 22
Fundamental properties of measures (or “size”):
Left Continuity: This is a trivial consequence of the definition!
Let then
Monday, March 19, 2012
Primbs, MS&E345 22
Fundamental properties of measures (or “size”):
Left Continuity: This is a trivial consequence of the definition!
Let then
Monday, March 19, 2012
Primbs, MS&E345 22
Fundamental properties of measures (or “size”):
Left Continuity: This is a trivial consequence of the definition!
Right Continuity: Depends on boundedness!
Let then
Monday, March 19, 2012
Primbs, MS&E345 22
Fundamental properties of measures (or “size”):
Left Continuity: This is a trivial consequence of the definition!
Right Continuity: Depends on boundedness!
Let then
Let and for some ithen
Monday, March 19, 2012
Primbs, MS&E345 22
Fundamental properties of measures (or “size”):
Left Continuity: This is a trivial consequence of the definition!
Right Continuity: Depends on boundedness!
(Here is why we need boundedness. Consider then )
Let then
Let and for some ithen
Monday, March 19, 2012
Primbs, MS&E345 23
Now we can define the integral.
Monday, March 19, 2012
Primbs, MS&E345 23
Now we can define the integral.
Simple Functions:
Monday, March 19, 2012
Primbs, MS&E345 23
Now we can define the integral.
Simple Functions:
Monday, March 19, 2012
Primbs, MS&E345 23
Now we can define the integral.
Simple Functions:
Positive Functions:
Monday, March 19, 2012
Primbs, MS&E345 23
Now we can define the integral.
Since positive measurable functions can be written as the increasing pointwise limit of simple functions, we define the integral as
Simple Functions:
Positive Functions:
Monday, March 19, 2012
Primbs, MS&E345 23
Now we can define the integral.
Since positive measurable functions can be written as the increasing pointwise limit of simple functions, we define the integral as
Simple Functions:
Positive Functions:
For a general measurable function write
Monday, March 19, 2012
Primbs, MS&E345 23
Now we can define the integral.
Since positive measurable functions can be written as the increasing pointwise limit of simple functions, we define the integral as
Simple Functions:
Positive Functions:
For a general measurable function write
Monday, March 19, 2012
Primbs, MS&E345 24
Monday, March 19, 2012
Primbs, MS&E345 24
How should I think about
Monday, March 19, 2012
Primbs, MS&E345 24
A
f
How should I think about
Monday, March 19, 2012
Primbs, MS&E345 24
This picture says everything!!!!
A
f
How should I think about
Monday, March 19, 2012
Primbs, MS&E345 24
This picture says everything!!!!
A
f
This is a set! and the integral is measuring the “size” of this set!
How should I think about
Monday, March 19, 2012
Primbs, MS&E345 24
This picture says everything!!!!
A
f
This is a set! and the integral is measuring the “size” of this set!
Integrals are like measures! They measure the size of a set. We just describe that set by a function.
How should I think about
Monday, March 19, 2012
Primbs, MS&E345 24
This picture says everything!!!!
A
f
This is a set! and the integral is measuring the “size” of this set!
Integrals are like measures! They measure the size of a set. We just describe that set by a function.
Therefore, integrals should satisfy the properties of measures.
How should I think about
Monday, March 19, 2012
Primbs, MS&E345 25
Monday, March 19, 2012
Primbs, MS&E345 25
This leads us to another important principle...
Monday, March 19, 2012
Primbs, MS&E345 25
Measures and integrals are different descriptions of of the same concept. Namely, “size”.
This leads us to another important principle...
Monday, March 19, 2012
Primbs, MS&E345 25
Measures and integrals are different descriptions of of the same concept. Namely, “size”.
Therefore, they should satisfy the same properties!!
This leads us to another important principle...
Monday, March 19, 2012
Primbs, MS&E345 25
Measures and integrals are different descriptions of of the same concept. Namely, “size”.
Therefore, they should satisfy the same properties!!
This leads us to another important principle...
Lebesgue defined the integral so that this would be true!
Monday, March 19, 2012
Primbs, MS&E345 26
Monday, March 19, 2012
Primbs, MS&E345 26
Measures are: Integrals are:
Monday, March 19, 2012
Primbs, MS&E345 26
Left Continuous
Measures are: Integrals are:
Monday, March 19, 2012
Primbs, MS&E345 26
Left Continuous
Measures are: Integrals are:
Monday, March 19, 2012
Primbs, MS&E345 26
Left Continuous Left Continuous
Measures are: Integrals are:
Monday, March 19, 2012
Primbs, MS&E345 26
Left Continuous Left Continuous
Measures are: Integrals are:
Monday, March 19, 2012
Primbs, MS&E345 26
Left Continuous Left Continuous
Measures are: Integrals are:
(Monotone Convergence Thm.)
Monday, March 19, 2012
Primbs, MS&E345 26
Left Continuous Left Continuous
Bdd Right Cont.
Measures are: Integrals are:
(Monotone Convergence Thm.)
Monday, March 19, 2012
Primbs, MS&E345 26
Left Continuous Left Continuous
Bdd Right Cont.
Measures are: Integrals are:
and
(Monotone Convergence Thm.)
Monday, March 19, 2012
Primbs, MS&E345 26
Left Continuous Left Continuous
Bdd Right Cont. Bdd Right Cont.
Measures are: Integrals are:
and
(Monotone Convergence Thm.)
Monday, March 19, 2012
Primbs, MS&E345 26
Left Continuous Left Continuous
Bdd Right Cont. Bdd Right Cont.
Measures are: Integrals are:
and and
(Monotone Convergence Thm.)
Monday, March 19, 2012
Primbs, MS&E345 26
Left Continuous Left Continuous
Bdd Right Cont. Bdd Right Cont.
Measures are: Integrals are:
and and
(Monotone Convergence Thm.)
(Bounded Convergence Thm.)
Monday, March 19, 2012
Primbs, MS&E345 26
Left Continuous Left Continuous
Bdd Right Cont. Bdd Right Cont.
etc...
Measures are: Integrals are:
and and
(Monotone Convergence Thm.)
(Bounded Convergence Thm.)
Monday, March 19, 2012
Primbs, MS&E345 26
Left Continuous Left Continuous
Bdd Right Cont. Bdd Right Cont.
etc...
Measures are: Integrals are:
and and
(Monotone Convergence Thm.)
(Bounded Convergence Thm.)
etc...(Fatou’s Lemma, etc...)
Monday, March 19, 2012
Primbs, MS&E345 27
The Lp Spaces
Monday, March 19, 2012
Primbs, MS&E345 27
The Lp Spaces
A function is in Lp(Ω,F,µ) if
Monday, March 19, 2012
Primbs, MS&E345 27
The Lp Spaces
A function is in Lp(Ω,F,µ) if
The Lp spaces are Banach Spaces with norm:
Monday, March 19, 2012
Primbs, MS&E345 27
The Lp Spaces
A function is in Lp(Ω,F,µ) if
The Lp spaces are Banach Spaces with norm:
L2 is a Hilbert Space with inner product:
Monday, March 19, 2012
Primbs, MS&E345 28
Perspective
σ-Algebras
Measurable Functions
Measure and Integration
Radon-Nikodym Theorem
Riesz Representation Theorem
Measure Theory
Probability Theory
Monday, March 19, 2012
Primbs, MS&E345 28
Perspective
σ-Algebras
Measurable Functions
Measure and Integration
Radon-Nikodym Theorem
Riesz Representation Theorem
Measure Theory
Probability Theory
Monday, March 19, 2012
Primbs, MS&E345 29
Monday, March 19, 2012
Primbs, MS&E345 29
Given a Measurable Space (Ω,F),
Monday, March 19, 2012
Primbs, MS&E345 29
Given a Measurable Space (Ω,F),
There exist many measures on F.
Monday, March 19, 2012
Primbs, MS&E345 29
Given a Measurable Space (Ω,F),
There exist many measures on F.
If Ω is the real line, the standard measure is “length”.
Monday, March 19, 2012
Primbs, MS&E345 29
Given a Measurable Space (Ω,F),
There exist many measures on F.
If Ω is the real line, the standard measure is “length”. That is, the measure of each interval is its length. This is known as “Lebesgue measure”.
Monday, March 19, 2012
Primbs, MS&E345 29
Given a Measurable Space (Ω,F),
There exist many measures on F.
If Ω is the real line, the standard measure is “length”. That is, the measure of each interval is its length. This is known as “Lebesgue measure”.
The σ-algebra must contain intervals. The smallest σ-algebra that contains all open sets (and hence intervals) is call the “Borel” σ-algebra and is denoted B.
Monday, March 19, 2012
Primbs, MS&E345 29
Given a Measurable Space (Ω,F),
There exist many measures on F.
If Ω is the real line, the standard measure is “length”. That is, the measure of each interval is its length. This is known as “Lebesgue measure”.
The σ-algebra must contain intervals. The smallest σ-algebra that contains all open sets (and hence intervals) is call the “Borel” σ-algebra and is denoted B.
A course in real analysis will deal a lot with the measurable space .
Monday, March 19, 2012
Primbs, MS&E345 30
Monday, March 19, 2012
Primbs, MS&E345 30
Given a Measurable Space (Ω,F),
Monday, March 19, 2012
Primbs, MS&E345 30
Given a Measurable Space (Ω,F),
A measurable space combined with a measure is called a measure space. If we denote the measure by µ, we would write the triple: (Ω,F,µ).
Monday, March 19, 2012
Primbs, MS&E345 30
Given a Measurable Space (Ω,F),
A measurable space combined with a measure is called a measure space. If we denote the measure by µ, we would write the triple: (Ω,F,µ).
Given a measure space (Ω,F,µ), if we decide instead to use a different measure, say υ, then we call this a “change of measure”. (We should just call this using another measure!)
Monday, March 19, 2012
Primbs, MS&E345 30
Given a Measurable Space (Ω,F),
A measurable space combined with a measure is called a measure space. If we denote the measure by µ, we would write the triple: (Ω,F,µ).
Given a measure space (Ω,F,µ), if we decide instead to use a different measure, say υ, then we call this a “change of measure”. (We should just call this using another measure!)
Let µ and υ be two measures on (Ω,F), then
Monday, March 19, 2012
Primbs, MS&E345 30
Given a Measurable Space (Ω,F),
A measurable space combined with a measure is called a measure space. If we denote the measure by µ, we would write the triple: (Ω,F,µ).
Given a measure space (Ω,F,µ), if we decide instead to use a different measure, say υ, then we call this a “change of measure”. (We should just call this using another measure!)
Let µ and υ be two measures on (Ω,F), then
(Notation )
υ is “absolutely continuous” with respect to µ if
Monday, March 19, 2012
Primbs, MS&E345 30
Given a Measurable Space (Ω,F),
A measurable space combined with a measure is called a measure space. If we denote the measure by µ, we would write the triple: (Ω,F,µ).
Given a measure space (Ω,F,µ), if we decide instead to use a different measure, say υ, then we call this a “change of measure”. (We should just call this using another measure!)
Let µ and υ be two measures on (Ω,F), then
(Notation )
υ is “absolutely continuous” with respect to µ if
υ and µ are “equivalent” if
Monday, March 19, 2012
Primbs, MS&E345 31
The Radon-Nikodym Theorem
Monday, March 19, 2012
Primbs, MS&E345 31
The Radon-Nikodym Theorem
If υ<<µ then υ is actually the integral of a function wrt µ.
Monday, March 19, 2012
Primbs, MS&E345 31
The Radon-Nikodym Theorem
If υ<<µ then υ is actually the integral of a function wrt µ.
Monday, March 19, 2012
Primbs, MS&E345 31
The Radon-Nikodym Theorem
If υ<<µ then υ is actually the integral of a function wrt µ.
Monday, March 19, 2012
Primbs, MS&E345 31
The Radon-Nikodym Theorem
If υ<<µ then υ is actually the integral of a function wrt µ.
Monday, March 19, 2012
Primbs, MS&E345 31
The Radon-Nikodym Theorem
If υ<<µ then υ is actually the integral of a function wrt µ.
Monday, March 19, 2012
Primbs, MS&E345 31
The Radon-Nikodym Theorem
If υ<<µ then υ is actually the integral of a function wrt µ.
g is known as the Radon-Nikodym derivative and denoted:
Monday, March 19, 2012
Primbs, MS&E345 32
The Radon-Nikodym Theorem
If υ<<µ then υ is actually the integral of a function wrt µ.
Monday, March 19, 2012
Primbs, MS&E345 32
The Radon-Nikodym Theorem
If υ<<µ then υ is actually the integral of a function wrt µ.
Idea of proof: Create the function through its superlevel sets
Monday, March 19, 2012
Primbs, MS&E345 32
The Radon-Nikodym Theorem
If υ<<µ then υ is actually the integral of a function wrt µ.
Consider the set function (this is actually a signed measure)
Idea of proof: Create the function through its superlevel sets
Monday, March 19, 2012
Primbs, MS&E345 32
The Radon-Nikodym Theorem
If υ<<µ then υ is actually the integral of a function wrt µ.
Consider the set function (this is actually a signed measure)
Idea of proof: Create the function through its superlevel sets
Choose and let be the largest set such that for all
(You must prove such an Aα exists.)
Monday, March 19, 2012
Primbs, MS&E345 32
The Radon-Nikodym Theorem
If υ<<µ then υ is actually the integral of a function wrt µ.
Consider the set function (this is actually a signed measure)
Then is the α-superlevel set of g.
Idea of proof: Create the function through its superlevel sets
Choose and let be the largest set such that for all
(You must prove such an Aα exists.)
Monday, March 19, 2012
Primbs, MS&E345 32
The Radon-Nikodym Theorem
If υ<<µ then υ is actually the integral of a function wrt µ.
Consider the set function (this is actually a signed measure)
Then is the α-superlevel set of g.
Idea of proof: Create the function through its superlevel sets
Choose and let be the largest set such that for all
(You must prove such an Aα exists.)
Now, given superlevel sets, we can construct a function by:
Monday, March 19, 2012
Primbs, MS&E345 33
Perspective
σ-Algebras
Measurable Functions
Measure and Integration
Radon-Nikodym Theorem
Riesz Representation Theorem
Measure Theory
Probability Theory
Monday, March 19, 2012
Primbs, MS&E345 33
Perspective
σ-Algebras
Measurable Functions
Measure and Integration
Radon-Nikodym Theorem
Riesz Representation Theorem
Measure Theory
Probability Theory
Monday, March 19, 2012
Primbs, MS&E345 34
The Riesz Representation Theorem:
Monday, March 19, 2012
Primbs, MS&E345 34
The Riesz Representation Theorem:
All continuous linear functionals on Lp are given by integration against a function with
Monday, March 19, 2012
Primbs, MS&E345 34
The Riesz Representation Theorem:
All continuous linear functionals on Lp are given by integration against a function with
That is, let be a cts. linear functional.
Then:
Monday, March 19, 2012
Primbs, MS&E345 34
The Riesz Representation Theorem:
All continuous linear functionals on Lp are given by integration against a function with
That is, let be a cts. linear functional.
Then:
Note, in L2 this becomes:
Monday, March 19, 2012
Primbs, MS&E345 35
The Riesz Representation Theorem:
All continuous linear functionals on Lp are given by integration against a function with
Monday, March 19, 2012
Primbs, MS&E345 35
The Riesz Representation Theorem:
All continuous linear functionals on Lp are given by integration against a function with
What is the idea behind the proof:
Monday, March 19, 2012
Primbs, MS&E345 35
The Riesz Representation Theorem:
All continuous linear functionals on Lp are given by integration against a function with
What is the idea behind the proof:
Linearity allows you to break things into building blocks, operate on them, then add them all together.
Monday, March 19, 2012
Primbs, MS&E345 35
The Riesz Representation Theorem:
All continuous linear functionals on Lp are given by integration against a function with
What is the idea behind the proof:
Linearity allows you to break things into building blocks, operate on them, then add them all together.
What are the building blocks of measurable functions.
Monday, March 19, 2012
Primbs, MS&E345 35
The Riesz Representation Theorem:
All continuous linear functionals on Lp are given by integration against a function with
What is the idea behind the proof:
Linearity allows you to break things into building blocks, operate on them, then add them all together.
What are the building blocks of measurable functions.
Indicator functions! Of course!
Monday, March 19, 2012
Primbs, MS&E345 35
The Riesz Representation Theorem:
All continuous linear functionals on Lp are given by integration against a function with
What is the idea behind the proof:
Linearity allows you to break things into building blocks, operate on them, then add them all together.
What are the building blocks of measurable functions.
Indicator functions! Of course!
Let’s define a set valued function from indicator functions:
Monday, March 19, 2012
Primbs, MS&E345 36
The Riesz Representation Theorem:
All continuous linear functionals on Lp are given by integration against a function with
Monday, March 19, 2012
Primbs, MS&E345 36
The Riesz Representation Theorem:
All continuous linear functionals on Lp are given by integration against a function with
A set valued function
Monday, March 19, 2012
Primbs, MS&E345 36
The Riesz Representation Theorem:
All continuous linear functionals on Lp are given by integration against a function with
A set valued function
How does L operate on simple functions
Monday, March 19, 2012
Primbs, MS&E345 36
The Riesz Representation Theorem:
All continuous linear functionals on Lp are given by integration against a function with
A set valued function
How does L operate on simple functions
This looks like an integral with υ the measure!
Monday, March 19, 2012
Primbs, MS&E345 36
The Riesz Representation Theorem:
All continuous linear functionals on Lp are given by integration against a function with
A set valued function
How does L operate on simple functions
This looks like an integral with υ the measure!
But, it is not too hard to show that υ is a (signed) measure. (countable additivity follows from continuity). Furthermore, υ<<µ. Radon-Nikodym then says dυ=gdµ.
Monday, March 19, 2012
Primbs, MS&E345 36
The Riesz Representation Theorem:
All continuous linear functionals on Lp are given by integration against a function with
A set valued function
How does L operate on simple functions
This looks like an integral with υ the measure!
But, it is not too hard to show that υ is a (signed) measure. (countable additivity follows from continuity). Furthermore, υ<<µ. Radon-Nikodym then says dυ=gdµ.
Monday, March 19, 2012
Primbs, MS&E345 37
The Riesz Representation Theorem:
All continuous linear functionals on Lp are given by integration against a function with
A set valued function
How does L operate on simple functions
This looks like an integral with υ the measure!
Monday, March 19, 2012
Primbs, MS&E345 37
The Riesz Representation Theorem:
All continuous linear functionals on Lp are given by integration against a function with
A set valued function
How does L operate on simple functions
This looks like an integral with υ the measure!
For measurable functions it follows from limits and continuity.
Monday, March 19, 2012
Primbs, MS&E345 37
The Riesz Representation Theorem:
All continuous linear functionals on Lp are given by integration against a function with
A set valued function
How does L operate on simple functions
This looks like an integral with υ the measure!
For measurable functions it follows from limits and continuity.
The details are left as an “easy” exercise for the reader...
Monday, March 19, 2012
Primbs, MS&E345 38
Monday, March 19, 2012
Primbs, MS&E345 38
The Riesz Representation Theorem and the Radon-Nikodym Theorem are basically equivalent. You can prove one from the other and vice-versa.
Monday, March 19, 2012
Primbs, MS&E345 38
The Riesz Representation Theorem and the Radon-Nikodym Theorem are basically equivalent. You can prove one from the other and vice-versa.
We will see the interplay between them in finance...
Monday, March 19, 2012
Primbs, MS&E345 38
The Riesz Representation Theorem and the Radon-Nikodym Theorem are basically equivalent. You can prove one from the other and vice-versa.
We will see the interplay between them in finance...
How does any of this relate to probability theory...
Monday, March 19, 2012
Primbs, MS&E345 39
Perspective
σ-Algebras
Measurable Functions
Measure and Integration
Radon-Nikodym Theorem
Riesz Representation Theorem
Measure Theory
Probability Theory
Monday, March 19, 2012
Primbs, MS&E345 39
Perspective
σ-Algebras
Measurable Functions
Measure and Integration
Radon-Nikodym Theorem
Riesz Representation Theorem
Measure Theory
Probability Theory
Monday, March 19, 2012
Primbs, MS&E345 40
Monday, March 19, 2012
Primbs, MS&E345 40
A probability measure P is a measure that satisfiesThat is, the measure of the whole space is 1.
Monday, March 19, 2012
Primbs, MS&E345 40
A random variable is a measurable function.
A probability measure P is a measure that satisfiesThat is, the measure of the whole space is 1.
Monday, March 19, 2012
Primbs, MS&E345 40
A random variable is a measurable function.
The expectation of a random variable is its integral:
A probability measure P is a measure that satisfiesThat is, the measure of the whole space is 1.
Monday, March 19, 2012
Primbs, MS&E345 40
A random variable is a measurable function.
The expectation of a random variable is its integral:
A density function is the Radon-Nikodym derivative wrt Lebesgue measure:
A probability measure P is a measure that satisfiesThat is, the measure of the whole space is 1.
Monday, March 19, 2012
Primbs, MS&E345 41
A probability measure P is a measure that satisfiesThat is, the measure of the whole space is 1.
Monday, March 19, 2012
Primbs, MS&E345 41
In finance we will talk about expectations with respect to different measures.
A probability measure P is a measure that satisfiesThat is, the measure of the whole space is 1.
Monday, March 19, 2012
Primbs, MS&E345 41
In finance we will talk about expectations with respect to different measures.
A probability measure P is a measure that satisfiesThat is, the measure of the whole space is 1.
Monday, March 19, 2012
Primbs, MS&E345 41
In finance we will talk about expectations with respect to different measures.
A probability measure P is a measure that satisfiesThat is, the measure of the whole space is 1.
Monday, March 19, 2012
Primbs, MS&E345 41
In finance we will talk about expectations with respect to different measures.
A probability measure P is a measure that satisfiesThat is, the measure of the whole space is 1.
And write expectations in terms of the different measures:
Monday, March 19, 2012
Primbs, MS&E345 41
In finance we will talk about expectations with respect to different measures.
A probability measure P is a measure that satisfiesThat is, the measure of the whole space is 1.
And write expectations in terms of the different measures:
Monday, March 19, 2012
Primbs, MS&E345 41
In finance we will talk about expectations with respect to different measures.
A probability measure P is a measure that satisfiesThat is, the measure of the whole space is 1.
where or
And write expectations in terms of the different measures:
Monday, March 19, 2012
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