# Definition:Event Space

## Definition

Let $\mathcal E$ be an experiment.

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

Each of the elements of $\Sigma$ are elements of the power set of $\Omega$, and are called events.

## Formal Definition

By definition, an experiment $\mathcal E$ has a probability space $\left({\Omega, \Sigma, \Pr}\right)$, which also by definition is a measure space.

Hence, again by definition, an **event space** $\Sigma$ is a sigma-algebra on $\Omega$.

Thus, an **event space** $\Sigma$ must fulfil the following requirements:

\((ES \ 1)\) | $:$ | Non-Empty: | \(\displaystyle \Sigma \) | \(\displaystyle \ne \) | \(\displaystyle \varnothing \) | |||

\((ES \ 2)\) | $:$ | Closure under Set Complement: | \(\displaystyle \forall A \in \Sigma:\) | \(\displaystyle \Omega \setminus A \) | \(\displaystyle \in \) | \(\displaystyle \Sigma \) | ||

\((ES \ 3)\) | $:$ | Closure under Countable Unions: | \(\displaystyle \forall A_1, A_2, \ldots \in \Sigma:\) | \(\displaystyle \bigcup_{i \mathop = 1}^\infty A_i \) | \(\displaystyle \in \) | \(\displaystyle \Sigma \) |

## Discrete Case

Let $\mathcal E$ be an experiment.

Let $\Omega$ be a discrete sample space of $\mathcal E$.

Then it is usual to take $\Sigma$ to be the power set $\mathcal P \left({\Omega}\right)$ of $\Omega$, that is, the set of all possible subsets of $\Omega$.

## Also denoted as

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

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

## Also see

## Sources

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