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.

Modern Probability Theory Foundations Measure Theory

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/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