Axiom:Kolmogorov Axioms

Definition

Let $\mathcal E$ be an experiment.

Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability measure on $\mathcal E$.

Then $\mathcal E$ can be defined as being a measure space $\left({\Omega, \Sigma, \Pr}\right)$, such that $\Pr \left({\Omega}\right) = 1$.

Thus $\Pr$ satisfies the Kolmogorov axioms:

Axioms

First Axiom

$\forall A \in \Sigma: 0 \le \Pr \left({A}\right) \le 1$

The probability of an event occurring is a real number between $0$ and $1$.

Second Axiom

$\Pr \left({\Omega}\right) = 1$

The probability of some elementary event occurring in the sample space is $1$.

Third Axiom

Let $A_1, A_2, \ldots$ be a countable (possibly countably infinite) sequence of pairwise disjoint events.

Then:

$\displaystyle \Pr \left({\bigcup_{i \ge 1} A_i}\right) = \sum_{i \ge 1} \Pr \left({A_i}\right)$

The probability of any one of countably many pairwise disjoint events occurring is the sum of the probabilities of the occurrence of each of the individual events.

As an elementary and easily-digested consequence of this, we have:

$\forall A, B \in \Sigma: A \cap B = \varnothing \implies \Pr \left({A \cup B}\right) = \Pr \left({A}\right) + \Pr \left({B}\right)$.

Also see

The axioms follow directly from the fact that $\left({\Omega, \Sigma, \Pr}\right)$ is a measure space.

Some sources include:

$\Pr \left({\varnothing}\right) = 0$

but this is strictly speaking not axiomatic as it can be deduced from the other axioms.

See Elementary Properties of Probability Measure.

Source of Name

This entry was named for Andrey Kolmogorov.