= Kolmogorov axioms
{c}
= Probability axioms
{synonym}
The <Kolmogorov axioms> describe a consistent way to assign <probabilities> to <events>. Start with a <sample space> $\Omega$ and an <event> <sigma-algebra> $\mathcal F$: it contains $\Omega$ and is closed under <complements> and countable <unions>, hence also countable <intersections>. The <probability measure> is a <function> $\mathbb P:\mathcal F\to\mathbb R$ satisfying
$$
\begin{aligned}
&\mathbb P(A)\geq0 &&\text{for every }A\in\mathcal F,\\
&\mathbb P(\Omega)=1,\\
&\mathbb P\left(\bigcup_{j=1}^{\infty}A_j\right)
=\sum_{j=1}^{\infty}\mathbb P(A_j)
&&\text{if }A_i\cap A_j=\varnothing\text{ for }i\ne j.
\end{aligned}
$$
The triple $(\Omega,\mathcal F,\mathbb P)$ is a <probability space>. The third axiom is <countable additivity>; the infinite <series> means the <limit of a sequence> of nonnegative partial sums. It applies to pairwise <disjoint events>, not to overlapping <events> and not to arbitrary uncountable <unions>.<Introduction to Probability, Lecture 1: Probability Models and Axioms by John Tsitsiklis and others>{ref}
The <sigma-algebra> specifies which questions have <probabilities>, while the axioms specify how those <probabilities> fit together. For a finite <sample space> one can take the entire <power set>; a continuous model commonly uses a <Borel sigma-algebra>. Nonnegativity and normalization do not require equally likely outcomes or <independence>: those are additional modelling choices.
For an exact six-outcome model, take $\Omega=\{1,2,3,4,5,6\}$, $\mathcal F=\mathcal P(\Omega)$, and assign $\mathbb P(\{k\})=k/21$. The total is one because $1+2+3+4+5+6=21$. Define the <probability> of an <event> by summing the weights of its outcomes. The <disjoint events> $A=\{1,2\}$, $B=\{3,4\}$ and $C=\{5,6\}$ form a <partition of a set> with
$$
\mathbb P(A)=\frac3{21},\qquad \mathbb P(B)=\frac7{21},\qquad
\mathbb P(C)=\frac{11}{21},\qquad
\mathbb P(A\cup B)=\frac{10}{21}=1-\mathbb P(C).
$$
This is a specified mathematical model, not measured die frequencies. The figure makes <probability> literal area: each region has the same height and a width proportional to its assigned mass. Regrouping <disjoint events> preserves their total area.
\Image[/probability-axioms-continuity-weighted-partition.png]
{title=Exact six-outcome model with masses k/21: grouping disjoint outcomes adds their probability areas}
Several familiar rules are consequences, not further axioms. <Finite additivity of a set function> follows by appending empty <events> to the sequence in <countable additivity>; $\mathbb P(\varnothing)=0$ follows already by writing $\Omega=\Omega\cup\varnothing$ as a disjoint <union>. The <complement rule of probability> gives $\mathbb P(A^c)=1-\mathbb P(A)$ and therefore $0\leq\mathbb P(A)\leq1$. Splitting larger <events> into a smaller <event> and a <set difference> shows $A\subseteq B\Rightarrow\mathbb P(A)\leq\mathbb P(B)$.
For overlapping <events>, the <inclusion-exclusion principle> corrects double counting:
$$
\mathbb P(D\cup E)=\mathbb P(D)+\mathbb P(E)-\mathbb P(D\cap E).
$$
In the six-outcome model, $D=\{2,3\}$ and $E=\{3,4\}$ give $5/21+7/21-3/21=9/21$, agreeing with the weight of $\{2,3,4\}$. This decomposition is useful because a complicated <event> can be assembled from simpler questions with known <probabilities>.
On a finite <sample space>, a family of pairwise <disjoint events> has only finitely many nonempty members, so <finite additivity of a set function> already handles every countable disjoint <union>. On an infinite <sample space>, the countable axiom additionally controls what is left beyond every finite stage. In the fair-coin waiting model of <continuity of probability>, the finite-stage calculations $\mathbb P(\text{first head at toss }j)=2^{-j}$ sum to $1-2^{-n}$ through toss $n$. <Countable additivity> makes the <probability> of a head at some finite toss equal to $\sum_{j\geq1}2^{-j}=1$; finite-stage sums alone do not justify that passage. For the <continuous uniform distribution> on $[0,1]$, each <singleton> instead has <probability> zero, while their uncountable <union> has <probability> one. Countable, rather than unrestricted, additivity accommodates both models.
Historically, Kolmogorov's 1933 book \i[<Grundbegriffe der Wahrscheinlichkeitsrechnung>] established the influential axiomatic framework connecting <probability theory> with <measure theory>. The three conditions above are the modern presentation on a <sigma-algebra>. Shafer and Vovk's historical study explains that the original presentation began with finite additivity on an <algebra of sets> and added continuity for decreasing <events> with empty <intersection>, which supplies the countable extension.<The Sources of Kolmogorov’s Grundbegriffe by Glenn Shafer and others>{ref}
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