# Events

An **experiment** is a procedure that can be repeated and whose observed outputs are uncertain.

The observed outputs of experiments are **outcomes**.

The set of possible outcomes is the **sample space** \(\Omega.\)

A subset of outcomes \(A\subseteq\Omega\) is called an **event**, and an event occurs if, when an experiment is performed, the outcome \(\omega\in\Omega\) satisfies \(\omega\in A.\)

The **null event** \(\varnothing\) and sample space \(\Omega\) are both events. The null event will never occur and the sample space event will always occur.

The singleton subsets of \(\Omega\) which are subsets that contain exactly one outcome from \(\Omega\) are called the **elementary events** of \(\Omega.\)

Events \(A\) and \(B\) are **disjoint events** if and only if \(A \cap B = \varnothing \,\, \forall \, A \neq B.\)

The set of all subsets of \(\Omega\) is the power set \(\mathcal{P}(\Omega).\) All events are subsets of \(\Omega\) however not all subsets of \(\mathcal{P}(\Omega)\) are necessarily events. Hence, the set of all possible events \(\mathcal{F}\) is a subset of \(\mathcal{P}(\Omega),\) \(\mathcal{F}\subseteq\mathcal{P}(\Omega).\)

If the set of events \(\mathcal{F}\) has the following properties:

- if events \(A,B\in\mathcal{F}\) then \(A\cup B\in\mathcal{F}\)
- if event \(A\in\mathcal{F}\) then \(\overline{A}\in\mathcal{F}\)
- the null event is in \(\mathcal{F},\) \(\varnothing\in\mathcal{F}\)

then \(\mathcal{F}\) is an **algebra** of sets. An algebra is closed under finite unions and finite intersections.

An algebra \(\mathcal{F}\) is a **\(\sigma\)-algebra** if it satisfies the following properies:

- if events \(E_1,E_2,\ldots\in\mathcal{F}\) then

- if event \(A\in\mathcal{F}\) then \(\overline{A}\in\mathcal{F}\)
- the null event is in \(\mathcal{F},\) \(\varnothing\in\mathcal{F}.\)

The elements in a \(\sigma\)-algebra are **measurable subsets** of the set it is defined on. A pair set containing a set and a \(\sigma\)-algebra of its measurable subsets is called a **measurable space**.

Any experiment has an associated measurable space \(\left(\Omega,\mathcal{F}\right)\) where \(\Omega\) is the set of all possible outcomes and \(\mathcal{F}\) is a \(\sigma\)-algebra of subsets of \(\Omega\) which contains all possible events.