Equivalence of Definitions of Sigma-Ring

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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

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


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.

$\blacksquare$


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In particular: Add equivalence to Definition 3
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