Equivalence of Definitions of Sigma-Ring

Definition 1 implies Definition 2
Let $\text {SR}$ be a ring of sets which is closed under countable unions.

We have:

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.

Definition 2 implies Definition 1
Let $\text {SR}$ be a system of sets such that:

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.