Definition:Event Space

Context
Probability Theory.

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.