# 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**.

## Sources

- 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $1$: Events and probabilities: $1.2$: Outcomes and events - 1991: Roger B. Myerson:
*Game Theory*... (previous) ... (next): $1.2$ Basic Concepts of Decision Theory