# Definition:Probability Measure

## Contents

## Definition

Let $\mathcal E$ be an experiment.

Let $\Omega$ be the sample space on $\mathcal E$, and let $\Sigma$ be the event space of $\mathcal E$.

A **probability measure on $\mathcal E$** is a mapping $\Pr: \Sigma \to \R$ which fulfils the Kolmogorov axioms.

## Kolmogorov Axioms

The Kolmogorov axioms are as follows:

### First Axiom

- $\forall A \in \Sigma: 0 \le \map \Pr A \le 1$

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

### Second Axiom

- $\map \Pr \Omega = 1$

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:

- $\displaystyle \map \Pr {\bigcup_{i \mathop \ge 1} A_i} = \sum_{i \mathop \ge 1} \map \Pr {A_i}$

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.

## Notes

From the definition of event space, we already have that $\varnothing \in \Sigma$ and $\Omega \in \Sigma$.

If $\mathcal E$ is defined as being a measure space $\left({\Omega, \Sigma, \Pr}\right)$, then $\Pr$ is a measure on $\mathcal E$ such that $\Pr \left({\Omega}\right) = 1$.

## Also known as

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

## Also see

## Sources

- 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $\S 1.3$: Probabilities - 1994: Martin J. Osborne and Ariel Rubinstein:
*A Course in Game Theory*... (previous) ... (next): $1.7$: Terminology and Notation - 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $4.2$