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

 Sana 18.12.2017 Hajmi 445 b. • ## We owe modern probability theory to the work of Andrey Nikolaevich Kolmogorov • ## 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.  • ## 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 • ## 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 • ## The probability of an event is a non-negative real number. P is always finite. • ## The probability that some elementary event (the event that contains only a single outcome) in the entire sample space will occur is 1. • ## Any countable sequence of pairwise disjoint events, E₁,E₂, ... satisfies P(E₁∪E₂∪...)=∑_{i=1}^{∞}P(E_{i}) • ## Monotonicity

• P(A)≤P(B) if A⊆B

• P(∅)=0
• ## The numeric bound

• It follows that 0≤P(E)≤1 for all E∈F • ## Therefore monotonicity and P(∅)=0 are proven. • ## Probability that internet works or we've paid for classes =.95 • ## http://en.wikipedia.org/wiki/Probability_axioms Do'stlaringiz bilan baham:

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