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Let $X$ be a nonempty set. We say $\mathcal{A} \subseteq \mathcal{P}(X)$ is an algebra if it is closed under finite union and complement. We say the same $\mathcal{A}$ is a $\sigma$-algebra if it is closed under countable union.

We say $\mathcal{R} \subseteq \mathcal{P}(X)$ is an ring if it is closed under finite union and set difference. We say the same $\mathcal{R}$ is a $\sigma$-ring if it is closed under countable union.


I am wondering if there is any connection between these terminologies and correspondent ones in abstract algebra. I think I have read somewhere that it is related to Boolean ring (I may be completely wrong), but I am afraid to sit down and write down details without if I am heading to a right direction. Any insightful comments or links (not for experts) will be appreciated.

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See also this question. –  Math Gems Jan 28 '13 at 20:41
    
Thanks for the link! –  GYC Jan 28 '13 at 22:04
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up vote 3 down vote accepted

Let $\mathcal R$ be a collection of sets which is closed under symmetric differences and intersections. Note that $\emptyset \in \mathcal R$. Then $\mathcal R$ is a commutative ring (in the algebraic sense) with addition defined by symmetric difference and multiplication defined by intersection, the zero of $\mathcal R$ is $\emptyset$. If we take the union of all the elements of $\mathcal R$ to be in $\mathcal R$ then it has a multiplicative identity. Note that for any $A \in \mathcal R$ we have $A+A=A \oplus A=\emptyset$ and $ A \cdot A=A \cap A=A$. It's a straightforward exercise to verify that the operations play nice and distribute as we would like. This sort of ring has a special name, it's what's called a boolean ring. In fact all such boolean rings arise in this manner by the stone representation theorem.

So there is some correspondence between the notations, don't ask me why some people call them fields. The $\sigma$ that gets thrown in front is a general way to refer to some sort of countable condition. If you continuing studying measure theory you'll run into the concept of a $\sigma$-finite measure space and in topology a $\sigma$-compact space.

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