# 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