# Axiom:Kolmogorov Axioms

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## Definition

Let $\EE$ be an experiment.

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability measure on $\mathcal E$.

Then $\EE$ can be defined as being a measure space $\struct {\Omega, \Sigma, \Pr}$, such that $\map \Pr \Omega = 1$.

Thus $\Pr$ satisfies the **Kolmogorov axioms**:

## Axioms

\((1)\) | $:$ | \(\displaystyle \forall A \in \Sigma:\) | \(\displaystyle 0 \) | \(\displaystyle \le \) | \(\displaystyle \map \Pr A \le 1 \) | The probability of an event occurring is a real number between $0$ and $1$ | ||

\((2)\) | $:$ | \(\displaystyle \map \Pr \Omega \) | \(\displaystyle = \) | \(\displaystyle 1 \) | The probability of some elementary event occurring in the sample space is $1$ | |||

\((3)\) | $:$ | \(\displaystyle \map \Pr {\bigcup_{i \mathop \ge 1} A_i} \) | \(\displaystyle = \) | \(\displaystyle \sum_{i \mathop \ge 1} \map \Pr {A_i} \) | where $\set {A_1, A_2, \ldots}$ is a countable (possibly countably infinite) set of pairwise disjoint events | |||

That is, 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:

- $\forall A, B \in \Sigma: A \cap B = \O \implies \map \Pr {A \cup B} = \map \Pr A + \map \Pr B$

## Also defined as

Some sources include:

- $\map \Pr \O = 0$

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

## Also see

- Definition:Measure Space: the
**Kolmogorov axioms**follow directly from the fact that $\struct {\Omega, \Sigma, \Pr}$ is an example of such.

## Source of Name

This entry was named for Andrey Nikolaevich Kolmogorov.