σ-algebra

Definition · applies to sets · also called sigma-algebra, σ-field

Undergraduate

A σ-algebra is a collection of the subsets of a set that can be measured, such as the events whose probabilities are defined. If a set can be measured, then so can everything outside it, and so can every combination of countably many measurable sets.

Graduate

Consider a set $X$ and a collection $\mathcal{F}$ of subsets of $X$. The collection $\mathcal{F}$ is a σ-algebra on $X$ if

1. $X$ is in $\mathcal{F}$,
2. the complement of every set in $\mathcal{F}$ is in $\mathcal{F}$, and
3. the union of every sequence of sets in $\mathcal{F}$ is in $\mathcal{F}$.

Examples

Examples

• $\setdef{\emptyset, \setdef{1}, \setdef{2, 3}, \setdef{1, 2, 3}}$, on $X = \setdef{1, 2, 3}$ Undergraduate
  The collection $\setdef{\emptyset, \setdef{1}, \setdef{2, 3}, \setdef{1, 2, 3}}$ is a σ-algebra on $\setdef{1, 2, 3}$, because it contains the complement of each of its sets, and every union of them.
• All subsets of a set $X$ (its power set) Undergraduate
  The power set of $X$ is a σ-algebra on $X$, because it contains every subset of $X$, so it is closed under every set operation.
• The Borel sets of $\reals$ Graduate
  The Borel sets of $\reals$ form a σ-algebra: the smallest σ-algebra that contains the open sets.

Non-examples

Things that may be mistaken for examples.

• The open sets of $\reals$ (the unions of open intervals) Undergraduate
  The complement of $(0, 1)$ is not open.
• $\setdef{\emptyset, \setdef{1}, \setdef{1, 2}}$, on $X = \setdef{1, 2}$ Undergraduate
  The collection $\setdef{\emptyset, \setdef{1}, \setdef{1, 2}}$ is not a σ-algebra on $\setdef{1, 2}$, because it does not contain $\setdef{2}$, the complement of $\setdef{1}$.
• The subsets of $\reals$ that are finite or have a finite complement Graduate
  The finite and cofinite subsets of $\reals$ do not form a σ-algebra. They are closed under complements and finite unions, but not countable unions: the union of the one-point sets $\setdef{n}$, for $n \in \nats$, is $\nats$, which is neither finite nor cofinite.

Properties

• A σ-algebra also contains $\emptyset$ and is closed under countable intersections, by De Morgan’s laws.
• The open sets of a topology generate a σ-algebra, the Borel σ-algebra: the smallest σ-algebra that contains every open set.

Notes

In probability, the sets in the σ-algebra are called events, and the “σ” refers to countable unions (sums).

Distinctions involving this term

• Topology vs. σ-algebra: Topologies and σ-algebras are both collections of subsets that are closed under some operations. A topology is closed under arbitrary unions and finite intersections, and describes nearness. A σ-algebra is closed under complements and countable unions, and describes what can be measured.

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