What properties must a collection of events have to be a sigma-field?

Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

sample space for a pair of dice — probability theory Sample space for a pair of dice.

In the context of probability theory, a collection of subsets $M$ of a sample space $S$ is defined as a $σ$ -field if it satisfies the following three properties:

Containment of the sample space: The sample space $S$ must be an element of $M$ ( $S \in M$ ).
Closure under complementation: If a set $A$ is in $M$ , then its complement $A^{c}$ must also be in $M$ .
Closure under countable unions: If a sequence of sets $A_{1}, A_{2}, \dots$ are all elements of $M$ , then their union ( $A_{1} \cup A_{2} \cup \dots$ ) must also be in $M$ .

These properties ensure that the collection is robust enough to allow for the consistent assignment of probabilities to events. From these axioms, it can be proven that a $σ$ -field also contains the empty set and is closed under countable intersections, as the latter can be expressed via De Morgan’s laws.1