# Definition:Sigma-Ring

## Definition

### Definition 1

A $\sigma$-ring is a ring of sets which is closed under countable unions.

That is, a ring of sets $\mathcal R$ is a $\sigma$-ring if and only if:

$\displaystyle A_1, A_2, \ldots \in \mathcal R \implies \bigcup_{n \mathop = 1}^\infty A_n \in \mathcal R$

### Definition 2

A $\sigma$-ring $\mathcal R$ is a system of sets with the following properties:

 $(SR \, 1)$ $:$ Empty Set: $\displaystyle \varnothing \in \mathcal R$ $(SR \, 2)$ $:$ Closure under Set Difference: $\displaystyle \forall A, B \in \mathcal R:$ $\displaystyle A \setminus B \in \mathcal R$ $(SR \, 3)$ $:$ Closure under Countable Unions: $\displaystyle \forall A_n \in \mathcal R: n = 1, 2, \ldots:$ $\displaystyle \bigcup_{n \mathop = 1}^\infty A_n \in \mathcal R$

### Definition 3

A $\sigma$-ring $\mathcal R$ is a system of sets with the following properties:

 $(SR \, 1')$ $:$ Empty Set: $\displaystyle \varnothing \in \mathcal R$ $(SR \, 2')$ $:$ Closure under Set Difference: $\displaystyle \forall A, B \in \mathcal R:$ $\displaystyle A \setminus B \in \mathcal R$ $(SR \, 3')$ $:$ Closure under Countable Disjoint Unions: $\displaystyle \forall A_n \in \mathcal R: n = 1, 2, \ldots:$ $\displaystyle \bigsqcup_{n \mathop = 1}^\infty A_n \in \mathcal R$

## Linguistic Note

The $\sigma$ in $\sigma$-ring is the Greek letter sigma which equates to the letter s.

$\sigma$ stands for for somme, which is French for union.