# Ring of Sets

写这篇主要是因为我看到有些 measure 的定义用到了 $\delta$-ring.

# Ring of Sets

Ring 的定义可以参考 Elementary Algebraic Structures

非常遗憾的是，**ring of sets 其实并不严格满足 ring 的定义** ()。按 Ring of Sets vs Ring in Universal Algebra 的讨论：

A “ring of sets” should really be called a “distributive lattice of sets.”

**Definition:** In measure theory, a nonempty family of sets $\mathcal{R}$ is called a ring (of sets) if it is closed under $\cup$ and $-$. That is, $\forall$ set $A,B \in R$,

- $A \cup B \in \mathcal{R}$
- $A - B \in \mathcal{R}$

# Ring of sets / $\sigma$-ring / $\delta$-ring

但是蛋疼的是，后续的 $\sigma$-ring 和 $\delta$-ring 都是在这个不严格的定义上引申出来的：

**Definition:** A $\sigma$-ring is a ring of sets which is closed under countable unions, i.e.

**Definition:** A $\delta$-ring is a ring of sets which is closed under countable intersections, i.e.

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