Different kings of Class
- Semi-Ring Definition: A class
of subsets such that where
- Ring Definition: A class
of subsets such that- First definition:
, - Second definition:
, - Third definition:
,
- First definition:
- Algebra Definition
- Any class of subsets of X that is a ring, and contains X
- Sigma Ring(Algebra) Definition: A class
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
Sigma-ring also means it is closed under countable intersections. Which means that both
- Monotone class: For any Monotone sequence in , its limit is in this class.
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
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
The proof is relatively difficult. 直观的思路是,不妨我们在