# 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 $\Sigma$ is a $\sigma$-ring if and only if:

$\ds A_1, A_2, \ldots \in \Sigma \implies \bigcup_{n \mathop = 1}^\infty A_n \in \Sigma$

### Definition 2

Let $\Sigma$ be a system of sets.

$\Sigma$ is a $\sigma$-ring if and only if $\Sigma$ satisfies the $\sigma$-ring axioms:

 $(\text {SR} 1)$ $:$ Empty Set: $\ds \O \in \Sigma$ $(\text {SR} 2)$ $:$ Closure under Set Difference: $\ds \forall A, B \in \Sigma:$ $\ds A \setminus B \in \Sigma$ $(\text {SR} 3)$ $:$ 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 3

Let $\Sigma$ be a system of sets.

$\Sigma$ is a $\sigma$-ring if and only if $\Sigma$ satisfies the $\sigma$-ring axioms:

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

## 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.