measuretheory demystified

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Measure Theory in a Lecture

Monday, March 19, 2012

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Perspective

σ-Algebras

Measurable Functions

Measure and Integration

Radon-Nikodym Theorem

Riesz Representation Theorem

Measure Theory

Probability Theory


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A little perspective:


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Riemann did this...


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Riemann did this...

...


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Lebesgue did this...


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Why is this better????


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Why is this better????


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Why is this better???? Consider the function:


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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


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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!


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Measure Theory begins from this simple motivation.


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If you remember a couple of simple principles, measure and integration theory becomes quite intuitive.


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The first principle is this...


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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.


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Perspective

σ-Algebras





Measure Theory

Probability Theory


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Measure theory is about measuring the size of things.


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In math we measure the size of sets.

Set A


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In math we measure the size of sets.

Set AHow big is the set A?


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“Size” should have the following property:


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The size of the union of disjoint sets should equal the sum of the sizes of the individual sets.


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Makes sense.... but there is a problem with this...


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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.


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If the size of a singleton is 0 then the size of [0,1] is 0.


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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!


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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.


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Sigma Algebras (σ-algebra)


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Sigma Algebras (σ-algebra)Consider a set Ω.


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Let F be a collection of subsets of Ω.


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F is a σ-algebra if:


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1)


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1)

2)


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1)

2)

3)


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1)

2)

3)

4)


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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)


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Perspective

σ-Algebras





Measure Theory

Probability Theory


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Algebras are what we need for super/sub-level sets of a vector space of indicator functions.


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Algebras are what we need for super/sub-level sets of a vector space of indicator functions.

Superlevel Sets


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Superlevel Sets


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Superlevel Sets

Simple Functions


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Superlevel Sets

Simple FunctionsSimple functions are finite linear combinations of of indicator functions.


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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.


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Given a Measurable Space (Ω,F)


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We say that a function: is measurable with respect to F if:


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This simply says that a function is measurable with respect to the σ-algebra if all its superlevel sets are in the σ-algebra.


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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?


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Another equivalent way to think of measurable functions is as the pointwise limit of simple functions.


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In fact, if a measurable function is non-negative, we can say it is the increasing pointwise limit of simple functions.


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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...


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Intuition behind measurable functions.


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They are “constant on the sets in the sigma algebra.”


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What do measurable functions look like?


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What do measurable functions look like?Yes


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What do measurable functions look like?Yes Yes


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What do measurable functions look like?Yes Yes No


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Intuitively speaking, Measurable functions are constant on the sets in the σ-alg.


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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.


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“Information” and σ-Algebras.


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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.


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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?


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determine about ω? The best possible I can do is to say that with


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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


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Perspective

σ-Algebras





Measure Theory

Probability Theory


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Measures and Integration


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A measurable space (Ω,F) defines the sets can be measured.


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Now we actually have to measure them...


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What are the properties that “size” should satisfy.


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What are the properties that “size” should satisfy.

If you think about it long enough, there are really only two...


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Definition of a measure:


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Definition of a measure:Given a Measurable Space (Ω,F),


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A measure is a functionsatisfying two properties:


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1)


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1)

2)


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1)

2)

(2) is known as “countable additivity”. However, I think of it as “linearity and left continuity for sets”.


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Fundamental properties of measures (or “size”):


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Left Continuity: This is a trivial consequence of the definition!


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Let then


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Right Continuity: Depends on boundedness!

Let then


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Let then

Let and for some ithen


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(Here is why we need boundedness. Consider then )

Let then

Let and for some ithen


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Now we can define the integral.


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Simple Functions:


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Simple Functions:

Positive Functions:


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Since positive measurable functions can be written as the increasing pointwise limit of simple functions, we define the integral as

Simple Functions:

Positive Functions:


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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


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How should I think about


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A

f



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This picture says everything!!!!

A

f



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A

f

This is a set! and the integral is measuring the “size” of this set!



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A

f


Integrals are like measures! They measure the size of a set. We just describe that set by a function.



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A

f


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.



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This leads us to another important principle...


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Measures and integrals are different descriptions of of the same concept. Namely, “size”.



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Therefore, they should satisfy the same properties!!



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Therefore, they should satisfy the same properties!!


Lebesgue defined the integral so that this would be true!


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Measures are: Integrals are:


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Left Continuous



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Left Continuous Left Continuous



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(Monotone Convergence Thm.)


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Bdd Right Cont.




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Bdd Right Cont.


and



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Bdd Right Cont. Bdd Right Cont.


and



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and and



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and and


(Bounded Convergence Thm.)


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etc...


and and




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etc...


and and



etc...(Fatou’s Lemma, etc...)


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The Lp Spaces


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The Lp Spaces

A function is in Lp(Ω,F,µ) if


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The Lp Spaces


The Lp spaces are Banach Spaces with norm:


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The Lp Spaces


The Lp spaces are Banach Spaces with norm:

L2 is a Hilbert Space with inner product:


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Perspective

σ-Algebras





Measure Theory

Probability Theory


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Given a Measurable Space (Ω,F),


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There exist many measures on F.


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If Ω is the real line, the standard measure is “length”.


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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”.


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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.


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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 .


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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).


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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!)


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Let µ and υ be two measures on (Ω,F), then


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(Notation )

υ is “absolutely continuous” with respect to µ if


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(Notation )

υ is “absolutely continuous” with respect to µ if

υ and µ are “equivalent” if


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The Radon-Nikodym Theorem


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If υ<<µ then υ is actually the integral of a function wrt µ.


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g is known as the Radon-Nikodym derivative and denoted:


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Idea of proof: Create the function through its superlevel sets


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Consider the set function (this is actually a signed measure)



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Choose and let be the largest set such that for all

(You must prove such an Aα exists.)


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Then is the α-superlevel set of g.





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Then is the α-superlevel set of g.




Now, given superlevel sets, we can construct a function by:


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Perspective

σ-Algebras





Measure Theory

Probability Theory


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The Riesz Representation Theorem:


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All continuous linear functionals on Lp are given by integration against a function with


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That is, let be a cts. linear functional.

Then:


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That is, let be a cts. linear functional.

Then:

Note, in L2 this becomes:


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What is the idea behind the proof:


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Linearity allows you to break things into building blocks, operate on them, then add them all together.


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What are the building blocks of measurable functions.


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Indicator functions! Of course!


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Indicator functions! Of course!

Let’s define a set valued function from indicator functions:


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A set valued function


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How does L operate on simple functions


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This looks like an integral with υ the measure!


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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.


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For measurable functions it follows from limits and continuity.


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For measurable functions it follows from limits and continuity.

The details are left as an “easy” exercise for the reader...


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The Riesz Representation Theorem and the Radon-Nikodym Theorem are basically equivalent. You can prove one from the other and vice-versa.


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We will see the interplay between them in finance...


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We will see the interplay between them in finance...

How does any of this relate to probability theory...


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Perspective

σ-Algebras





Measure Theory

Probability Theory


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A probability measure P is a measure that satisfiesThat is, the measure of the whole space is 1.


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A random variable is a measurable function.



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The expectation of a random variable is its integral:



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The expectation of a random variable is its integral:

A density function is the Radon-Nikodym derivative wrt Lebesgue measure:



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In finance we will talk about expectations with respect to different measures.



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And write expectations in terms of the different measures:


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where or

And write expectations in terms of the different measures:


measuretheory demystified

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