Definition:Sigma-Algebra

Definition

Definition 1

Let $X$ be a set.

A $\sigma$-algebra $\mathcal R$ over $X$ is a system of subsets of $X$ with the following properties:

\((SA \, 1)\)	$:$	Unit:		\(\displaystyle X \in \mathcal R \)
\((SA \, 2)\)	$:$	Closure under Complement:	\(\displaystyle \forall A \in \mathcal R:\)	\(\displaystyle \complement_X \left({A}\right) \in \mathcal R \)
\((SA \, 3)\)	$:$	Closure under Countable Unions:	\(\displaystyle \forall A_n \in \mathcal R: n = 1, 2, \ldots:\)	\(\displaystyle \bigcup_{n \mathop = 1}^\infty A_n \in \mathcal R \)

Definition 2

Let $X$ be a set.

A $\sigma$-algebra $\mathcal R$ over $X$ is a system of subsets of $X$ with the following properties:

\((SA \, 1')\)	$:$	Unit:		\(\displaystyle X \in \mathcal R \)
\((SA \, 2')\)	$:$	Closure under Complement:	\(\displaystyle \forall A \in \mathcal R:\)	\(\displaystyle \complement_X \left({A}\right) \in \mathcal R \)
\((SA \, 3')\)	$:$	Closure under Countable Disjoint Unions:	\(\displaystyle \forall A_n \in \mathcal R: n = 1, 2, \ldots:\)	\(\displaystyle \bigsqcup_{n \mathop = 1}^\infty A_n \in \mathcal R \)

Definition 3

A $\sigma$-algebra $\mathcal R$ is a $\sigma$-ring with a unit.

Definition 4

Let $X$ be a set.

A $\sigma$-algebra $\mathcal R$ over $X$ is an algebra of sets which is closed under countable unions.

Also known as

This is also seen as $\sigma$-algebra, from $\sigma$ being the Greek letter sigma.

Also see

• Results about $\sigma$-algebras can be found here.

• Equivalence of Definitions of Sigma-Algebra
• $\sigma$-Algebra as Magma of Sets, proving that $\sigma$-algebras instantiate the general concept of a magma of sets.

• Sigma-Algebra is Delta-Algebra
• Delta-Algebra is Sigma-Algebra

Linguistic Note

The $\sigma$ in $\sigma$-algebra is the Greek letter sigma which equates to the letter s.

$\sigma$ stands for for somme, which is French for union.