The Derivation of Modern Probability Theory from Measure Theory By Jeffrey Carrington

The Derivation of Modern Probability Theory from Measure Theory

• By Jeffrey Carrington

Introduction

• 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.)

• 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.

Measure Theory

• What is measure?

• Encountered as the “length” of a ruler, the “area” of a room and the “volume” of a cup.

• Involves the assigning of a number to a set.

• Certain properties are intrinsic to measure:

• “Measure” is nonnegative.
• “Measure” can be +∞
• 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.)

• Metric Space

• 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

• The probability of an event is a non-negative real number. P is always finite.

Second Probability Axiom

• 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

• Any countable sequence of pairwise disjoint events, E₁,E₂, ... satisfies P(E₁∪E₂∪...)=∑_{i=1}^{∞}P(E_{i})

Consequences of Kolmogorov Axioms

• Monotonicity

• P(A)≤P(B) if A⊆B

• The probability of the empty set

• P(∅)=0

• The numeric bound

• It follows that 0≤P(E)≤1 for all E∈F

Consequences (cont.)

• 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

• 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/measure-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