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 $

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 $\Omega$.

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.

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 $\EE$.

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

\((\text {ES} 1)\)	$:$	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:	\(\ds \forall A_1, A_2, \ldots \in \Sigma:\)	\(\ds \bigcup_{i \mathop = 1}^\infty A_i \)	\(\ds \in \)	\(\ds \Sigma \)