DefinitionProved

Kolmogorov's axioms of probability

Statement

A probability space is a triple $(\Omega, \mathcal{F}, \mathbb{P})$ consisting of a sample space $\Omega$ , a $\sigma$ -algebra $\mathcal{F}$ of subsets of $\Omega$ (called events), and a function $\mathbb{P} : \mathcal{F} \to \mathbb{R}$ satisfying three axioms: (1) non-negativity: $\mathbb{P}(A) \ge 0$ for every $A \in \mathcal{F}$ ; (2) normalization: $\mathbb{P}(\Omega) = 1$ ; and (3) countable additivity: for every sequence $A_1, A_2, \dots \in \mathcal{F}$ of pairwise disjoint events ( $A_i \cap A_j = \varnothing$ for $i \ne j$ ), $\mathbb{P}\left(\bigcup_{i=1}^{\infty} A_i\right) = \sum_{i=1}^{\infty} \mathbb{P}(A_i)$ .

Why is it true?

Kolmogorov's axioms treat probability just like mass or area distributed across the space of possible outcomes $\Omega$ : every event has non-negative weight, the total weight of all possibilities is $1$ , and the weight of a union of non-overlapping pieces — even countably infinitely many — is simply the sum of their individual weights.

Proof sketch

From the three axioms all standard rules of probability follow immediately: taking $A_1 = \Omega$ and $A_i = \varnothing$ for $i \ge 2$ gives $1 = \mathbb{P}(\Omega) = \mathbb{P}(\Omega) + \sum_{i=2}^{\infty} \mathbb{P}(\varnothing)$ , hence $\mathbb{P}(\varnothing) = 0$ . Finite additivity follows by setting $A_i = \varnothing$ for $i > n$ . Since $A$ and $A^c$ are disjoint with $A \cup A^c = \Omega$ , we get $\mathbb{P}(A^c) = 1 - \mathbb{P}(A)$ , monotonicity $\mathbb{P}(A) \le \mathbb{P}(B)$ when $A \subseteq B$ , and continuity of probability along monotone sequences of events via countable additivity.

Stated by

stated 1933Andrey Kolmogorov

Proved by

proved 1933Andrey Kolmogorov

Topics that use this theorem

Events and probability

Related theorems

Bayes' theorem Chebyshev's inequality Weak law of large numbers Strong law of large numbers

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

Andrey Kolmogorov (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung
Andrey Kolmogorov (1950). Foundations of the Theory of Probability