Definition:Event Space
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Context
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.
Some sources use $\mathcal F$ or a script version of $F$ to denote an event space.
Event Space as a Sigma-Algebra
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:
- $\Sigma \ne \varnothing$, that is, an event space can not be empty.
- If $A \in \Sigma$, then $\Omega - A \in \Sigma$, that is, the complement of $A$ relative to $\Omega$ is also in $\Sigma$.
- If $A_1, A_2, \ldots \in \Sigma$, then $\displaystyle \bigcup_{i=1}^\infty A_i \in \Sigma$, that is, the union of any countable collection of elements of $\Sigma$ is also in $\Sigma$.
Discrete Case
If $\Omega$ is a discrete sample space, 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 $\Omega$.
From Power Set of Sample Space is an Event Space it can be seen that this is a valid approach.
Further Elementary Properties
See Elementary Properties of Event Space for some further results.
