Definition:Sigma-Algebra
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Definition
Definition 1
Let $X$ be a set.
A $\sigma$-algebra $\Sigma$ over $X$ is a system of subsets of $X$ with the following properties:
| \((\text {SA} 1)\) | $:$ | Unit: | \(\displaystyle X \in \Sigma \) | |||||
| \((\text {SA} 2)\) | $:$ | Closure under Complement: | \(\displaystyle \forall A \in \Sigma:\) | \(\displaystyle \relcomp X A \in \Sigma \) | ||||
| \((\text {SA} 3)\) | $:$ | Closure under Countable Unions: | \(\displaystyle \forall A_n \in \Sigma: n = 1, 2, \ldots:\) | \(\displaystyle \bigcup_{n \mathop = 1}^\infty A_n \in \Sigma \) |
Definition 2
Let $X$ be a set.
A $\sigma$-algebra $\Sigma$ over $X$ is a system of subsets of $X$ with the following properties:
| \((\text {SA} 1')\) | $:$ | Unit: | \(\displaystyle X \in \Sigma \) | |||||
| \((\text {SA} 2')\) | $:$ | Closure under Complement: | \(\displaystyle \forall A \in \Sigma:\) | \(\displaystyle \relcomp X A \in \Sigma \) | ||||
| \((\text {SA} 3')\) | $:$ | Closure under Countable Disjoint Unions: | \(\displaystyle \forall A_n \in \Sigma: n = 1, 2, \ldots:\) | \(\displaystyle \bigsqcup_{n \mathop = 1}^\infty A_n \in \Sigma \) |
Definition 3
A $\sigma$-algebra $\Sigma$ is a $\sigma$-ring with a unit.
Definition 4
Let $X$ be a set.
A $\sigma$-algebra $\Sigma$ 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
- 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.
- Definition:Measurable Space: the resulting structure $\struct {X, \Sigma}$
- 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, and also summe, which is German for union.