集合类 | Icarus's Net

Different kings of Class

Semi-Ring Definition: A class $\varphi$ of subsets such that
- $\varnothing \in \varphi$
- $A,\, B \in \varphi$ $\Rightarrow$ $A \cap B \in \varphi$
- $A,\, B \in \varphi$ $\Rightarrow$ $A-B = \cup_{i = 1}^{n} E_i$ where $E_i \in \varphi$
Ring Definition: A class $\mathcal{R}$ of subsets such that
- First definition: $A,\, B \in \mathcal{R}$ $\Rightarrow$ $A \cap B \in \mathcal{R}$, $A \Delta B \in \mathcal{R}$
- Second definition: $A,\, B \in \mathcal{R}$ $\Rightarrow$ $A \cap B \in \mathcal{R}$, $A \cup B \in \mathcal{R}$
- Third definition: $A,\, B \in \mathcal{R}$ $\Rightarrow$ $A \cup B \in \mathcal{R}$, $A - B \in \mathcal{R}$
Algebra Definition
- Any class of subsets of X that is a ring, and contains X
Sigma Ring(Algebra) Definition: A class $\mathcal{R}$ of subsets such that
- The Ring closed under countable unions. Same for Sigma Field(which is also called Borel Field, sigma-algebra)

Semi-ring 直观例子就是$(a,b]$ 组成的集合。以上三个条件显然满足。然而上述定义的$(a,b]$组成的集合并不是一种Ring, 原因在于该类对于$A \Delta B$ 不封闭。更加有趣的是，对于Ring of sets, 同样可以用代数的Ring 来理解。

Sigma-ring also means it is closed under countable intersections. Which means that both $E_i $ and $ E_i$ are in the set of $\mathcal{R}$

Monotone class: For any Monotone sequence in , its limit is in this class.

Obviously, a ring is a monotone class. Also, if a monotone class is a ring, it is naturally a sigma ring.

除了Semi Ring以外的上述结构，我们不妨定义为z-class, 对于同一种z-class 都有如下性质（十分重要的性质）：任意同一类z-class 的交都是z-class. 有了这个结论以后，可以立即证明有包含给定类（关于X的子集）的最小z-class. 我们称这个集合是 the z-class generated by X.

Semi Ring 可以通过一定方式来生成一个 Ring, 具体而言，有如下的定理

The ring generated by $\varphi$ can be expressed:\[ E = \cup_{k = 1}^{n} A_k\] where $A_k$ are finite disjoint sets of $\varphi$

定理的证明是技巧性的。具体来说，只要证明如上定义的交和difference结果都可以表达为上述定义结果即可。

如果我们已经有了一个Ring, 由它生成的 monotone class 就是这个ring生成的sigma ring. 这是一个非常重要的结论。

Monotone class theorem for sets

Let $G$ be an algebra of sets and define $M(G)$ to be the smallest monotone class containing $G$ Then $M(G)$ is precisely the 𝜎-algebra generated by $M(G)$

The proof is relatively difficult. 直观的思路是，不妨我们在$M(G)$的每一个元素$F$的基础上构造集合（也就是满足$E-F$, $F-E$, $E \cup F$, $E \in M(G)$ 都在$M(G)$中的集合$E$），组成新的class $\mathcal{Z}(F)$。这样的构造说明对于任何在$M(G)$中的元素，都可以找到另一个跟$F$形成Ring关系的元素。主要说明对任意的$F$，都可以成立$\mathcal{Z}(F) = M(G)$ 就可以了。证明非常巧妙的地方，就是利用了上述定义的对称性，以及monotone class 的性质。