# Equivalence of Definitions of Sigma-Ring

## Theorem

The following definitions of the concept of $\sigma$-ring are equivalent:

### 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

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

 $(\text {SR} 1)$ $:$ 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

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

 $(\text {SR} 1')$ $:$ 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: $\ds \forall A_n \in \Sigma: n = 1, 2, \ldots:$ $\ds \bigsqcup_{n \mathop = 1}^\infty A_n \in \Sigma$

## Proof

### Definition 1 implies Definition 2

Let $\text {SR}$ be a ring of sets which is closed under countable unions.

We have:

 $(\text {RS} 1_2)$ $:$ Empty Set: $\ds \O \in \text {SR}$ $(\text {RS} 2_2)$ $:$ Closure under Set Difference: $\ds \forall A, B \in \text {SR}:$ $\ds A \setminus B \in \text {SR}$

which are exactly $\text {SR} 1$ and $\text {SR} 2$.

Then as $\text {SR}$ is closed under countable unions:

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

and so $\text {SR} 3$ is fulfilled.

$\Box$

### Definition 2 implies Definition 1

Let $\text {SR}$ be a system of sets such that:

 $(\text {SR} 1)$ $:$ $\ds \O \in \text {SR}$ $(\text {SR} 2)$ $:$ $\ds \forall A, B \in \text {SR}:$ $\ds A \setminus B \in \text {SR}$ $(\text {SR} 3)$ $:$ $\ds \forall A_n \in \text {SR}: n = 1, 2, \ldots:$ $\ds \bigcup_{n \mathop = 1}^\infty A_n \in \text {SR}$

As noted above, $\text {SR} 1$ and $\text {SR} 2$ are exactly $\text {RS} 1_2$ and $\text {RS} 2_2$.

Let $A, B \in \text {SR}$.

Let $A_1 = A, A_2 = B$ and $A_n = \O$ for all $n = 3, 4, \ldots$

Then:

$\ds \forall A_n \in \text {SR}: n = 1, 2, \ldots: \bigcup_{n \mathop = 1}^\infty A_n = A \cup B \in \text {SR}$

Thus criterion $(\text {RS} 3_2)$ is fulfilled.

So $\text {SR}$ is a ring of sets which is closed under countable unions.

$\blacksquare$