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


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The Derivation of Modern Probability Theory from Measure Theory


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



Consequences of Kolmogorov Axioms



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




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