An event is the mathematical form of a question about an experiment. A coin is tossed, a card is drawn, a point is sampled, a random path is observed; in every case probability can only assign numbers after the question has been translated into a set of outcomes. The theory of events explains which questions are allowed, how logical operations on questions become set operations, and why countable limits of questions remain inside probability theory.

The main danger is that verbal questions feel more flexible than measurable sets. In a finite model this causes no trouble, because every subset can be assigned a probability. In an uncountable model, such as a uniformly chosen point of $[0,1]$, asking for probabilities of all subsets is incompatible with the usual rules of length and countable additivity. Events are therefore not arbitrary sentences; they are measurable subsets chosen as part of the model.

[example: A Die Roll as a Set of Outcomes] Let $\Omega=\{1,2,3,4,5,6\}$ with the uniform probability measure, so $\mathbb P(\{k\})=1/6$ for each $k\in\Omega$. The question "is the outcome even?" is represented by the event \begin{align*} A=\{\omega\in\Omega:\omega\text{ is even}\}=\{2,4,6\}\subset\Omega. \end{align*} The three singleton events $\{2\}$, $\{4\}$, and $\{6\}$ are pairwise disjoint, and \begin{align*} A=\{2\}\cup\{4\}\cup\{6\}. \end{align*} By finite additivity for disjoint events, \begin{align*} \mathbb P(A) &=\mathbb P(\{2\}\cup\{4\}\cup\{6\})\\ &=\mathbb P(\{2\})+\mathbb P(\{4\})+\mathbb P(\{6\})\\ &=\frac{1}{6}+\frac{1}{6}+\frac{1}{6}\\ &=\frac{3}{6}\\ &=\frac{1}{2}. \end{align*} Thus the probabilistic question is answered by first identifying the set of outcomes where the phrase "even outcome" is true, and then measuring that set. [/example]

This finite example is deliberately simple, but it already contains the whole pattern. We need a sample space of possible outcomes, a class of subsets that count as observable questions, and a probability rule on that class. Before naming events themselves, we should build the structure in which such questions are allowed to live.

## Event Structures

The event collection cannot be an arbitrary list of favourite questions. If a model can answer a question, it should also answer its negation; if it can answer countably many questions, it should answer whether at least one of them occurs. These requirements lead to the closure rules of a sigma-algebra.

[definition: Sigma-Algebra of Events] Let $\Omega$ be a set. A sigma-algebra of events on $\Omega$ is a collection $\mathcal F\subset\mathcal P(\Omega)$ such that \begin{align*} &\Omega\in\mathcal F,\\ &A\in\mathcal F \implies A^c\in\mathcal F,\\ &A_1,A_2,\dots\in\mathcal F \implies \bigcup_{n=1}^{\infty}A_n\in\mathcal F. \end{align*} [/definition]

The [complement rule](/theorems/4970) represents the logical word "not", and the countable union rule represents countable uses of "or". These closure rules also give countable intersections by taking complements. Once this allowable class of questions is fixed, the next ingredient is the numerical rule that measures them.

[definition: Probability Space] A probability space is a triple $(\Omega,\mathcal F,\mathbb P)$ consisting of a sample space $\Omega$, a sigma-algebra $\mathcal F$ on $\Omega$, and a function \begin{align*} \mathbb P:\mathcal F&\to[0,1] \end{align*} such that $\mathbb P(\Omega)=1$ and, whenever $A_1,A_2,\dots\in\mathcal F$ are pairwise disjoint, \begin{align*} \mathbb P\left(\bigcup_{n=1}^{\infty}A_n\right)=\sum_{n=1}^{\infty}\mathbb P(A_n). \end{align*} [/definition]

This definition separates three choices that are often blurred: possible outcomes, observable questions, and assigned probabilities. With the ambient probability space in place, the primary definition can now be stated in its compact form.

## Definition

The definition of an event answers the smallest modelling question: once a probability space has been chosen, what kind of object is a probabilistic question? It should be something to which $\mathbb P$ can assign a number. This forces events to be members of the sigma-algebra, not merely subsets named in prose.

[definition: Event] Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space. An event is an element $A\in\mathcal F$. [/definition]

The definition is short because the structure has been pushed into $\mathcal F$. In a countable model, individual outcomes can be useful building blocks for events; when a singleton has been admitted into $\mathcal F$, it is often called an elementary event. Elementary events are decisive in finite probability, but they do not capture the full behaviour of continuous probability. This distinction motivates the first warning example after the definition. It shows why events cannot always be understood by adding the masses of individual outcomes.

[example: Singletons in a Continuous Model] Let $\Omega=[0,1]$, let $\mathcal F=\mathcal B([0,1])$, where $\mathcal B([0,1])$ denotes the Borel sigma-algebra on $[0,1]$, and let $\mathbb P$ be [Lebesgue measure](/page/Lebesgue%20Measure) restricted to $[0,1]$. For $a\in[0,1]$, the singleton is the degenerate interval \begin{align*} \{a\}=[a,a]. \end{align*} It is closed in $[0,1]$, hence it belongs to $\mathcal B([0,1])$, so it is an event. Since restricted Lebesgue measure agrees with interval length on intervals contained in $[0,1]$, \begin{align*} \mathbb P(\{a\}) &=\mathbb P([a,a])\\ &=a-a\\ &=0. \end{align*} For $0\le r<s\le 1$, the interval $[r,s]$ is also closed in $[0,1]$, hence is an event, and \begin{align*} \mathbb P([r,s]) &=s-r. \end{align*} Thus every individual point has probability zero, but an interval with distinct endpoints has positive probability equal to its length. [/example]

The continuous example forces us to respect the sigma-algebra. It also prepares the algebraic viewpoint: events are sets, so the logic of probability is built from set operations. The next section develops that calculus because almost every probability computation begins by rewriting the target event.

## Algebra of Events

### Intersections and Unions

The same event can be described in many verbal ways, and the most useful description is often obtained by combining simpler events. To compute probabilities, we need reliable translations of "and", "or", and "not". The next definition starts with simultaneous occurrence, since overlap is the quantity that appears in conditional probability and independence.

[definition: Intersection of Events] Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space and let $A,B\in\mathcal F$. The intersection event of $A$ and $B$ is \begin{align*} A\cap B=\{\omega\in\Omega: \omega\in A \text{ and } \omega\in B\}. \end{align*} [/definition]