# Definition:Event Space

## Definition

Let $\EE$ be an experiment whose probability space is $\struct {\Omega, \Sigma, \Pr}$.

The event space of $\EE$ is usually denoted $\Sigma$ (Greek capital sigma), and is the set of all outcomes of $\EE$ which are interesting.

By definition, $\struct {\Omega, \Sigma}$ is a measurable space.

Hence the event space $\Sigma$ is a sigma-algebra on $\Omega$.

That is:

 $(\text {ES} 1)$ $:$ Non-Empty: $\ds \Sigma$ $\ds \ne$ $\ds \O$ $(\text {ES} 2)$ $:$ Closure under Set Complement: $\ds \forall A \in \Sigma:$ $\ds \Omega \setminus A$ $\ds \in$ $\ds \Sigma$ $(\text {ES} 3)$ $:$ Closure under Countable Unions: $\ds \forall A_1, A_2, \ldots \in \Sigma:$ $\ds \bigcup_{i \mathop = 1}^\infty A_i$ $\ds \in$ $\ds \Sigma$

### Discrete Event Space

Let $\EE$ be an experiment.

Let $\Omega$ be a discrete sample space of $\EE$.

Then it is commonplace to take $\Sigma$ to be the power set $\powerset \Omega$ of $\Omega$, that is, the set of all possible subsets of $\Omega$.

## Also denoted as

Some sources use $\FF$ or $\mathscr F$ to denote an event space.

In the field of decision theory, the symbol $\Xi$ can often be seen.

## Examples

### Arbitrary Event Space on 6-Sided Die

Let $\EE$ be the experiment of throwing a standard $6$-sided die.

The sample space of $\EE$ is $\Omega = \set {1, 2, 3, 4, 5, 6}$.

Let $\FF$ be the arbitrary set of subsets of $\Omega$ defined as:

$\FF = \set {\O, \set {1, 2}, \set {3, 4}, \set {5, 6}, \set {1, 2, 3, 4}, \set {3, 4, 5, 6}, \set {1, 2, 5, 6}, \Omega}$

Then $E$ is an event space of $\EE$.

## Also see

• Results about event spaces can be found here.