Definition:Event Space

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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:       \(\displaystyle \Sigma \)   \(\displaystyle \ne \)   \(\displaystyle \O \)             
\((\text {ES} 2)\)   $:$   Closure under Set Complement:      \(\displaystyle \forall A \in \Sigma:\)    \(\displaystyle \Omega \setminus A \)   \(\displaystyle \in \)   \(\displaystyle \Sigma \)             
\((\text {ES} 3)\)   $:$   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 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