Primbs, MS&E345 1 Measure Theory in a Lecture

Primbs, MS&E345 2 Perspective  -Algebras Measurable Functions Measure and Integration Radon-Nikodym Theorem Riesz Representation Theorem Measure Theory Probability Theory

Primbs, MS&E345 3 A little perspective: Riemann did this......

Primbs, MS&E345 4 A little perspective: Lebesgue did this...

Primbs, MS&E345 5 A little perspective: Lebesgue did this... Why is this better????

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 sublevel sets, we can integrate a lot of function!

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 sub-level sets.

Primbs, MS&E345 8 Perspective  -Algebras Measurable Functions Measure and Integration Radon-Nikodym Theorem Riesz Representation Theorem Measure Theory Probability Theory

Primbs, MS&E345 9 Measure theory is about measuring the size of things. In math we measure the size of sets. Set A How big is the set A?

Primbs, MS&E “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.

Primbs, MS&E Sigma Algebras (  -algebra) Consider a set  Let F be a collection of subsets of  F is a  -algebra if: An algebra of sets is closed under finite set operations. A  -algebra is closed under countable set operations. In mathematics,  often refers to “countable”. 1) 2) 3) 4)

Primbs, MS&E Perspective  -Algebras Measurable Functions Measure and Integration Radon-Nikodym Theorem Riesz Representation Theorem Measure Theory Probability Theory

Primbs, MS&E Algebras are what we need for sublevel sets of a vector space of indicator functions. Sublevel Sets

Primbs, MS&E Sublevel Sets  algebras let us take limits of these. That is more interesting!! Simple Functions Simple functions are finite linear combinations of of indicator functions.

Primbs, MS&E Measurable Functions Given a Measurable Space ( ,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  -algebra if all its sublevel sets are in the  -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?

Primbs, MS&E 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...

Primbs, MS&E Intuition behind measurable functions. They are “constant on the sets in the sigma algebra.” What do measurable functions look like? Yes No Intuitively speaking, Measurable functions are constant on the sets in the  -alg. More accurately, they are limits of functions are constant on the sets in the  - algebra. The  in  -algebra gives us this.

Primbs, MS&E “Information” and  -Algebras. Since measurable functions are “constant” on the  -algebra, if I am trying to determine information from a measurable function, the  -algebra determines the information that I can obtain. I measure What is the most information that I determine about  ?  -algebras determine the amount of information possible in a function. The best possible I can do is to say that with

Primbs, MS&E Perspective  -Algebras Measurable Functions Measure and Integration Radon-Nikodym Theorem Riesz Representation Theorem Measure Theory Probability Theory

Primbs, MS&E Measures and Integration A measurable space (  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...

Primbs, MS&E Definition of a measure: Given a Measurable Space ( ,F), A measure is a function satisfying two properties: 1) 2) (2) is known as “countable additivity”. However, I think of it as “linearity and left continuity for sets”.

Primbs, MS&E 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 i then

Primbs, MS&E 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

Primbs, MS&E 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 must satisfy the properties of measures. How should I think about

Primbs, MS&E Measures and integrals are different descriptions of of the same concept. Namely, “size”. Therefore, they must satisfy exactly the same properties!! This leads us to another important principle... Lebesgue defined the integral so that this would be true!

Primbs, MS&E Left Continuous Bdd Right Cont. etc... Measures are: Integrals are: and (Monotone Convergence Thm.) (Bounded Convergence Thm.) etc... (Fatou’s Lemma, etc...)

Primbs, MS&E The L p Spaces A function is in L p ( ,F,  ) if The L p spaces are Banach Spaces with norm: L 2 is a Hilbert Space with inner product:

Primbs, MS&E Perspective  -Algebras Measurable Functions Measure and Integration Radon-Nikodym Theorem Riesz Representation Theorem Measure Theory Probability Theory

Primbs, MS&E Given a Measurable Space ( ,F), There exist many measures on F. If  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  -algebra must contain intervals. The smallest  - algebra that contains all open sets (and hence intervals) is call the “Borel”  -algebra and is denoted B. A course in real analysis will deal a lot with the measurable space.

Primbs, MS&E Given a Measurable Space ( ,F), A measurable space combined with a measure is called a measure space. If we denote the measure by , we would write the triple: ( ,F  Given a measure space ( ,F  if we decide instead to use a different measure, say  then we call this a “change of measure”. (We should just call this using another measure!) Let  and  be two measures on ( ,F), then (Notation )  is “absolutely continuous” with respect to  if  and  are “equivalent” if

Primbs, MS&E The Radon-Nikodym Theorem If  <<  then  is actually the integral of a function wrt . g is known as the Radon- Nikodym derivative and denoted:

Primbs, MS&E The Radon-Nikodym Theorem If  <<  then  is actually the integral of a function wrt . Consider the set function (this is actually a signed measure) Then is the  -sublevel set of g. Idea of proof: Create the function through its sublevel sets Chooseand letbe the largest set such that for all (You must prove such an A  exists.) Now, given sublevel sets, we can construct a function by:

Primbs, MS&E Perspective  -Algebras Measurable Functions Measure and Integration Radon-Nikodym Theorem Riesz Representation Theorem Measure Theory Probability Theory

Primbs, MS&E The Riesz Representation Theorem: All continuous linear functionals on L p are given by integration against a function with That is, letbe a cts. linear functional. Then: Note, in L 2 this becomes:

Primbs, MS&E The Riesz Representation Theorem: All continuous linear functionals on L p 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:

Primbs, MS&E The Riesz Representation Theorem: All continuous linear functionals on L p 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  the measure! But, it is not too hard to show that  is a (signed) measure. (countable additivity follows from continuity). Furthermore,  <<  Radon-Nikodym then says d  =gd 

Primbs, MS&E The Riesz Representation Theorem: All continuous linear functionals on L p 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  the measure! For measurable functions it follows from limits and continuity. The details are left as an “easy” exercise for the reader...

Primbs, MS&E 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...

Primbs, MS&E Perspective  -Algebras Measurable Functions Measure and Integration Radon-Nikodym Theorem Riesz Representation Theorem Measure Theory Probability Theory

Primbs, MS&E 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 satisfies That is, the measure of the whole space is 1.

Primbs, MS&E In finance we will talk about expectations with respect to different measures. A probability measure P is a measure that satisfies That is, the measure of the whole space is 1. whereor And write expectations in terms of the different measures: