Measure Theory in a Lecture

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Measure Theory in a Lecture
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Perspective
s-Algebras
Measurable Functions
Measure Theory
Measure and Integration
Radon-Nikodym Theorem
Riesz Representation Theorem
Probability Theory
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A little perspective
Riemann did this...
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A little perspective
Lebesgue did this...
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A little perspective
Lebesgue did this...
Why is this better????
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A little perspective
Lebesgue did this...
Why is this better????
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.
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.
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Perspective
s-Algebras
Measurable Functions
Measure Theory
Measure and Integration
Radon-Nikodym Theorem
Riesz Representation Theorem
Probability Theory
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Measure theory is about measuring the size of
things.
How big is the set A?
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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.
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Sigma Algebras (s-algebra)
Consider a set W.
Let F be a collection of subsets of W.
An algebra of sets is closed under finite set
operations. A s-algebra is closed under
countable set operations. In mathematics, s
often refers to countable.
F is a s-algebra if
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Perspective
s-Algebras
Measurable Functions
Measure Theory
Measure and Integration
Radon-Nikodym Theorem
Riesz Representation Theorem
Probability Theory
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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
Simple Functions
Simple functions are finite linear combinations
of of indicator functions.
s-algebras let us take limits of these. That is
more interesting!!
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Measurable Functions
Given a Measurable Space (W,F)
This simply says that a function is measurable
with respect to the s-algebra if all its
superlevel sets are in the s-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.
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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Intuition behind measurable functions.
They are constant on the sets in the sigma
algebra.
What do measurable functions look like?
Yes
No
Yes
Intuitively speaking, Measurable functions are
constant on the sets in the s-alg.
More accurately, they are limits of functions are
constant on the sets in the s-algebra. The s in
s-algebra gives us this.
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Information and s-Algebras.
Since measurable functions are constant on the
s-algebra, if I am trying to determine
information from a measurable function, the
s-algebra determines the information that I can
obtain.
s-algebras determine the amount of information
possible in a function.
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Perspective
s-Algebras
Measurable Functions
Measure Theory
Measure and Integration
Radon-Nikodym Theorem
Riesz Representation Theorem
Probability Theory
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Measures and Integration
A measurable space (W,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...
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Definition of a measure
Given a Measurable Space (W,F),
(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)
Left Continuity This is a trivial consequence of
the definition!
Right Continuity Depends on boundedness!
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Now we can define the integral.
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This picture says everything!!!!
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...
Measures and integrals are different descriptions
of of the same concept. Namely, size.
Therefore, they should satisfy the same
properties!!
Lebesgue defined the integral so that this would
be true!
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Left Continuous
Left Continuous
(Monotone Convergence Thm.)
Bdd Right Cont.
Bdd Right Cont.
(Bounded Convergence Thm.)
etc...
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The Lp Spaces
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Perspective
s-Algebras
Measurable Functions
Measure Theory
Measure and Integration
Radon-Nikodym Theorem
Riesz Representation Theorem
Probability Theory
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Given a Measurable Space (W,F),
There exist many measures on F.
If W 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 s-algebra must contain intervals. The
smallest s-algebra that contains all open sets
(and hence intervals) is call the Borel
s-algebra and is denoted B.
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Given a Measurable Space (W,F),
A measurable space combined with a measure is
called a measure space. If we denote the measure
by m, we would write the triple (W,F,m).
Given a measure space (W,F,m), if we decide
instead to use a different measure, say u, then
we call this a change of measure. (We should
just call this using another measure!)
Let m and u be two measures on (W,F), then
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The Radon-Nikodym Theorem
If ultltm then u is actually the integral of a
function wrt m.
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The Radon-Nikodym Theorem
If ultltm then u is actually the integral of a
function wrt m.
Idea of proof Create the function through its
superlevel sets
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Perspective
s-Algebras
Measurable Functions
Measure Theory
Measure and Integration
Radon-Nikodym Theorem
Riesz Representation Theorem
Probability Theory
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The Riesz Representation Theorem
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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!
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The Riesz Representation Theorem
All continuous linear functionals on Lp are given
by integration against a function
with
But, it is not too hard to show that u is a
(signed) measure. (countable additivity follows
from continuity). Furthermore, ultltm.
Radon-Nikodym then says dugdm.
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The Riesz Representation Theorem
All continuous linear functionals on Lp are given
by integration against a function
with
This looks like an integral with u the measure!
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.

We will see the interplay between them in
finance...
How does any of this relate to probability
theory...
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Perspective
s-Algebras
Measurable Functions
Measure Theory
Measure and Integration
Radon-Nikodym Theorem
Riesz Representation Theorem
Probability Theory
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In finance we will talk about expectations with
respect to different measures.
And write expectations in terms of the different
measures