Equivalence of Definitions of Sigma-Ring

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

Definition 1 implies Definition 2
Let $\mathcal R$ be a ring of sets which is closed under countable unions.

We have:

which are exactly $SR \, 1$ and $SR \, 2$.

Then as $\mathcal R$ is closed under countable unions:


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

and so $SR \, 3$ is fulfilled.

Definition 2 implies Definition 1
Let $\mathcal R$ be a system of sets such that for all $A, B \in \mathcal R$:
 * $(SR \, 1): \quad \varnothing \in \mathcal R$
 * $(SR \, 2): \quad A \setminus B \in \mathcal R$
 * $(SR \, 3): \quad \displaystyle \forall A_n \in \mathcal R: n = 1, 2, \ldots: \bigcup_{n \mathop = 1}^\infty A_n \in \mathcal R$

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

Let $A, B \in \mathcal R$.

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

Then:
 * $\displaystyle \forall A_n \in \mathcal R: n = 1, 2, \ldots: \bigcup_{n \mathop = 1}^\infty A_n = A \cup B \in \mathcal R$

Thus criterion $(RS \, 3_2)$ is fulfilled.

So $\mathcal R$ is a ring of sets which is closed under countable unions.