Definition:Event Space

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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.


Formal Definition

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:

\((ES \ 1)\)   $:$   Non-Empty:       \(\displaystyle \Sigma \)   \(\displaystyle \ne \)   \(\displaystyle \varnothing \)             
\((ES \ 2)\)   $:$   Closure under Set Complement:      \(\displaystyle \forall A \in \Sigma:\)    \(\displaystyle \Omega \setminus A \)   \(\displaystyle \in \)   \(\displaystyle \Sigma \)             
\((ES \ 3)\)   $:$   Closure under Countable Unions:      \(\displaystyle \forall A_1, A_2, \ldots \in \Sigma:\)    \(\displaystyle \bigcup_{i \mathop = 1}^\infty A_i \)   \(\displaystyle \in \)   \(\displaystyle \Sigma \)             


Discrete Case

Let $\mathcal E$ be an experiment.

Let $\Omega$ be a discrete sample space of $\mathcal E$.


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 $\Omega$.


Also denoted as

Some sources use $\mathcal F$ or $\mathscr F$ to denote an event space.

In the field of decision theory, the symbol $\Xi$ can often be seen.


Also see


Sources