The Derivation of Modern Probability Theory from Measure Theory

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

2. 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

3. 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.

4. Modern Probability Theory Foundations Measure Theory

5. 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

6. 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.

7. Probability Axioms

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

9. 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.

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

11. 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

12. 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.

13. 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

14. 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