Definition:Sigma-Algebra
Contents
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.
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.
