Definition
The Kolmogorov axioms are the three rules Andrey Kolmogorov stated in his 1933 monograph Grundbegriffe der Wahrscheinlichkeitsrechnung that anchor modern probability theory:
- Probabilities are non-negative real numbers.
- The probability of the entire sample space is 1.
- For any countable collection of mutually exclusive events, the probability of their union equals the sum of their probabilities (countable additivity).
By recasting probability as a measure on a sigma-algebra of subsets of the sample space, Kolmogorov gave the subject a rigorous mathematical foundation that finally separated probability's mathematics from its philosophical interpretation.
Why it matters
How it works
In the Kolmogorov framework, a probability space is a triple (Ω, F, P): the sample space Ω, a sigma-algebra F of admissible events (subsets of Ω closed under complement and countable union), and a probability measure P assigning a number in [0, 1] to each event in F. The three axioms constrain P.
From this minimal scaffolding emerge conditional probability, independence, expected value, random variables, convergence theorems, and stochastic processes — the full toolkit of modern probability. The axioms themselves are simple; the implications fill libraries.