集合类

Different kings of Class

  • Semi-Ring Definition: A class φ of subsets such that
    • φ
    • A,Bφ ABφ
    • A,Bφ AB=i=1nEi where Eiφ
  • Ring Definition: A class R of subsets such that
    • First definition: A,BR ABR, AΔBR
    • Second definition: A,BR ABR, ABR
    • Third definition: A,BR ABR, ABR
  • Algebra Definition
    • Any class of subsets of X that is a ring, and contains X
  • Sigma Ring(Algebra) Definition: A class 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AΔB 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 Ei and Ei are in the set of 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 φ can be expressed:E=k=1nAk where Ak are finite disjoint sets of φ

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EF, FE, EF, EM(G) 都在M(G)中的集合E),组成新的class 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。