Download Measure Theory and Probability Theory

The Derivation of Modern Probability
Theory from Measure Theory
By Jeffrey Carrington
Introduction
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The question of “Will a specific event occur?”
has always been a concern of man.
Probability theory is concerned with the analysis
of this random phenomena and allows us to
quantify the likelihood of an event occuring.
We owe modern probability theory to the work
of Andrey Nikolaevich Kolmogorov
Introduction (cont.)
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Andrey Kolmogorov was a soviet
mathematician born in 1903.
He combined the idea of sample space with
measure theory and created the axiom system
for modern probability theory in 1933.
These axioms can be summarized by the
statement: Let (Ω,Ϝ,P) be a measure space with
P(Ω)=1. Then (Ω,Ϝ,P) is a probability space with
sample space Ω, event space F and probability
measure P.
Modern Probability Theory Foundations
Measure Theory
Measure Theory
What is measure?
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Encountered as the “length” of a ruler, the
“area” of a room and the “volume” of a cup.
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Involves the assigning of a number to a set.
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Certain properties are intrinsic to measure:
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“Measure” is nonnegative.
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“Measure” can be +∞
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If A is a subset of R, it can be written as
A=U_{n}A_{n} where the A_{n}'s are disjoint
non-empty subintervals of A
Measure Theory (cont.)
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Metric Space
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A set such that the concept of distance
between elements is defined. It is
represented as (S,d) where S is a set
and d is a metric such that d:SxS-->R. It
also has the properties of positivity,
symmetry, identity and triangle
inequality.
Probability Axioms
First Probability Axiom
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The probability of an event is a non-negative
real number. P is always finite.
Second Probability Axiom
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The probability that some elementary event
(the event that contains only a single outcome)
in the entire sample space will occur is 1.
Third Probability Axiom
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Any countable sequence of pairwise disjoint
events, E₁,E₂, ... satisfies
P(E₁∪E₂∪...)=∑_{i=1}^{∞}P(E_{i})
Consequences of Kolmogorov
Axioms
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Monotonicity
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The probability of the empty set
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P(A)≤P(B) if A⊆B
P(∅)=0
The numeric bound
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It follows that 0≤P(E)≤1 for all E∈F
Consequences (cont.)
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Let E₁=A and E₂=B/A, where A⊆B and
E_{i}=∅ for i≥3. By the third axiom
E₁∪E₂∪...=B and
P(A)+P(B\A)+∑_{i=3}^{∞}P(∅)=P(B).
Now if P(∅)>0 then by set theory definitions
we would obtain a contradiction.
Additionally P(A)≤P(B).
Therefore monotonicity and P(∅)=0 are
proven.
Example
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We attempt to register for class. We
successfully register if and only if the internet
works and we have paid for classes.
Probability (internet works)=.9, Probability(paid
for classes)=.6 and P(internet works and paid
for classes)=.55
Probability that internet works or we've paid for
classes =.95
Works Cited
Measure Theory Tutorial.
https://www.ee.washington.edu/techsite/papers/
documents/UWEETR-2006-0008.pdf
 An Introduction to Measure Theory
http://terrytao.files.wordpress.com/2011/01/meas
ure-book1.pdf
 The Theory of Measures and Integration, Eric
M. Vestrup
 http://en.wikipedia.org/wiki/Probability_theory
 http://en.wikipedia.org/wiki/Probability_axioms
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