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