Definition:Probability Measure

Context

Probability Theory.

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.

These are as follows:

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:

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

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 see Elementary Properties of Probability Measure for further immediate consequences of this definition.

Sources

• Geoffrey Grimmett: Probability: An Introduction (1986): $\S 1.3$