# 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.

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: | \(\displaystyle \forall A_n \in \mathcal R: n = 1, 2, \ldots:\) | \(\displaystyle \bigcup_{n \mathop = 1}^\infty A_n \in \mathcal R \) |

## Subcategories

This category has only the following subcategory.

## Pages in category "Sigma-Algebras"

The following 50 pages are in this category, out of 50 total.

### B

### C

- Cardinality of Infinite Sigma-Algebra is at Least Cardinality of Continuum
- Characterization of Euclidean Borel Sigma-Algebra
- Characterization of Extended Real Sigma-Algebra
- Characterization of Sigma-Algebra Generated by Collection of Mappings
- Closed Set Measurable in Borel Sigma-Algebra
- Condition on Equality of Generated Sigma-Algebras

### D

### E

### G

- Generated Sigma-Algebra by Generated Monotone Class
- Generated Sigma-Algebra by Generated Monotone Class/Corollary
- Generated Sigma-Algebra Contains Generated Dynkin System
- Generated Sigma-Algebra Contains Generated Sigma-Algebra of Subset
- Generated Sigma-Algebra Preserves Finiteness
- Generated Sigma-Algebra Preserves Subset
- Generator for Product Sigma-Algebra
- Generators for Extended Real Sigma-Algebra

### 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 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 Generated by Complements of Generators
- Sigma-Algebra is Delta-Algebra
- Sigma-Algebra is Dynkin System
- Sigma-Algebra is Monotone Class
- Sigma-Algebra of Countable Sets