Proof — Kolmogorov's axioms of probability
Statement
Probability is a measure on a σ-algebra of events. Three axioms: (1) P(E) ≥ 0 for any event E; (2) P(Ω) = 1 for the sample space; (3) P(⋃Eᵢ) = ∑P(Eᵢ) for countable disjoint events. This measure-theoretic foundation unifies discrete and continuous probability and underpins modern statistics, stochastic processes, and machine learning.
Formula
Probability as a measure on a measurable space
Equation
P(E) ≥ 0, P(Ω) = 1, P(⋃Eᵢ) = ∑P(Eᵢ)
Source
Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung, 1933
Domains
mathematics