Category:Sigma-Algebras
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This category contains results about Sigma-Algebras.
Definitions specific to this category can be found in Definitions/Sigma-Algebras.
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: | \(\ds \forall A_n \in \Sigma: n = 1, 2, \ldots:\) | \(\ds \bigcup_{n \mathop = 1}^\infty A_n \in \Sigma \) |
Subcategories
This category has the following 12 subcategories, out of 12 total.
B
- Borel Probability Spaces (empty)
C
E
- Examples of Sigma-Algebras (2 P)
F
M
P
- Product Sigma-Algebras (2 P)
S
T
- Trace Sigma-Algebras (8 P)
Pages in category "Sigma-Algebras"
The following 36 pages are in this category, out of 36 total.
C
D
E
I
P
S
- Sigma-Algebra as Magma of Sets
- Sigma-Algebra Closed under Countable Intersection
- Sigma-Algebra Closed under Finite Intersection
- Sigma-Algebra Closed under Finite Union
- Sigma-Algebra Closed under Intersection
- Sigma-Algebra Closed under Set Difference
- Sigma-Algebra Closed under Symmetric Difference
- Sigma-Algebra Closed under Union
- Sigma-Algebra Closed under Union/Corollary
- Sigma-Algebra Contains Empty Set
- Sigma-Algebra Contains Generated Sigma-Algebra of Subset
- Sigma-Algebra Extended by Single Set
- Sigma-Algebra is Delta-Algebra
- Sigma-Algebra is Dynkin System
- Sigma-Algebra is Monotone Class
- Sigma-Algebra of Countable Sets
- Stopped Sigma-Algebra is Sigma-Algebra