# 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.