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 $$\bigcup_{i=1}^\infty A_i \in \Sigma$$, 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.