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.

The set $$\Sigma$$ itself, by the nature of a probability space, is a sigma-algebra on $$\Omega$$.

That is, for $$\Sigma$$ actually to be an event space, it 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 collection of elements of $$\Sigma$$ is also in $$\Sigma$$.

It follows from Properties of Algebras of Sets, and that $$\Sigma$$ is a sigma-algebra, that both $$\varnothing \in \Sigma$$ and $$\Omega \in \Sigma$$.