# Definition:Probability Measure

## Definition

Let $\EE$ be an experiment.

### Definition 1

Let $\EE$ be defined as a measure space $\struct {\Omega, \Sigma, \Pr}$.

Then $\Pr$ is a measure on $\EE$ such that $\map \Pr \Omega = 1$.

### Definition 2

Let $\Omega$ be the sample space on $\EE$.

Let $\Sigma$ be the event space of $\mathcal E$.

A **probability measure on $\EE$** is a mapping $\Pr: \Sigma \to \R$ which fulfils the Kolmogorov 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 |

### Definition 3

Let $\Omega$ be the sample space on $\EE$.

Let $\Sigma$ be the event space of $\mathcal E$.

A **probability measure on $\EE$** is a mapping $\Pr: \Sigma \to \R$ which fulfils the following axioms:

\((\text I)\) | $:$ | \(\displaystyle \forall A \in \Sigma:\) | \(\displaystyle \map \Pr A \) | \(\displaystyle \ge \) | \(\displaystyle 0 \) | |||

\((\text {II})\) | $:$ | \(\displaystyle \map \Pr \Omega \) | \(\displaystyle = \) | \(\displaystyle 1 \) | ||||

\((\text {III})\) | $:$ | \(\displaystyle \forall A \in \Sigma:\) | \(\displaystyle \map \Pr A \) | \(\displaystyle = \) | \(\displaystyle \sum_{\bigcup \set e \mathop = A} \map \Pr {\set e} \) | where $e$ denotes the elementary events of $\EE$ |

### Definition 4

Let $\Omega$ be the sample space on $\EE$.

Let $\Sigma$ be the event space of $\mathcal E$.

A **probability measure on $\EE$** is an additive function $\Pr: \Sigma \to \R$ which fulfils the following axioms:

\((1)\) | $:$ | \(\displaystyle \forall A, B \in \Sigma: A \cap B = \O:\) | \(\displaystyle \map \Pr {A \cup B} \) | \(\displaystyle = \) | \(\displaystyle \map \Pr A + \map \Pr B \) | |||

\((2)\) | $:$ | \(\displaystyle \map \Pr \Omega \) | \(\displaystyle = \) | \(\displaystyle 1 \) |

## Also known as

In the context of probability theory, a **probability measure** is sometimes referred to as a **probability function**.

## Also see

- Definition:Event Space: $\O \in \Sigma$ and $\Omega \in \Sigma$

- Results about
**probability measures**can be found here.