Definition:Sigma-Algebra

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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.


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.