Definition:Sigma-Algebra
Definition
Definition 1
Let $X$ be a set.
Let $\Sigma$ be a system of subsets of $X$.
$\Sigma$ is a $\sigma$-algebra over $X$ if and only if $\Sigma$ satisfies the sigma-algebra axioms:
| \((\text {SA} 1)\) | $:$ | Unit: | \(\ds X \in \Sigma \) | ||||||
| \((\text {SA} 2)\) | $:$ | Closure under Complement: | \(\ds \forall A \in \Sigma:\) | \(\ds \relcomp X A \in \Sigma \) | |||||
| \((\text {SA} 3)\) | $:$ | Closure under Countable Unions: | \(\ds \forall A_n \in \Sigma: n = 1, 2, \ldots:\) | \(\ds \bigcup_{n \mathop = 1}^\infty A_n \in \Sigma \) |
Definition 2
Let $X$ be a set.
Let $\Sigma$ be a system of subsets of $X$.
$\Sigma$ is a $\sigma$-algebra over $X$ if and only if $\Sigma$ satisfies the sigma-algebra axioms:
| \((\text {SA} 1')\) | $:$ | Unit: | \(\ds X \in \Sigma \) | ||||||
| \((\text {SA} 2')\) | $:$ | Closure under Set Difference: | \(\ds \forall A, B \in \Sigma:\) | \(\ds A \setminus B \in \Sigma \) | |||||
| \((\text {SA} 3')\) | $:$ | Closure under Countable Disjoint Unions: | \(\ds \forall A_n \in \Sigma: n = 1, 2, \ldots:\) | \(\ds \bigsqcup_{n \mathop = 1}^\infty A_n \in \Sigma \) |
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.
Examples
Trivial $\sigma$-Algebra
Let $X$ be a set.
The trivial $\sigma$-algebra on $X$ is the $\sigma$-algebra defined as:
- $\set {\O, X}$
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.