Borel-Cantelli Lemma

ICTP Real Analysis

前言：该笔记开始于2020暑假的最后一个月。由于Royden这本书中已经将定理及其证明讲述得很详细，写这个笔记的目的主要是疏通脉络，同时挑选一些比较重要或者有代表性的定理证明。对于每一个部分的开头，我试图整理出一些主要的脉络和逻辑链。这样即使对于这些知识的构架变得生疏了以后，也能很快通过重新阅读笔记来捡起来 :)

更新(10/3/2020)：对于ICTP课程中一些更加一般化的测度论的结论，打算通过IMPA的课程以查漏补缺的方式填上。因此这个笔记会逐渐增添内容。

Part 0

sigma-algebra ${\textstyle \to }$ Borel set,

${\textstyle G_{\delta }}$ set, ${\textstyle F_{\sigma }}$ set.

(Prop) Every nonempty open set is the disjoint union of a countable collection of open intervals.

证明摘要：基于开集中的任意${\textstyle x}$，定义 ${\textstyle a_{x}:=\inf\{z|(z,x)\subseteq O\}}$ 和 ${\textstyle a_{x}:=\sup\{y|(x,y)\subseteq O\}}$ 且${\textstyle I_{x}:=(a_{x},b_{x})}$. 易证${\textstyle a_{x},b_{x}\notin O}$. 根据有理数在实数中的稠密性，可在 ${\textstyle I_{x}}$ 和有理数${\textstyle k\in I_{x}}$ 之间建立一一对应关系。显然${\textstyle \{I_{x}\}}$ 不相交，且因其与有理数的一一对应对应关系可数。得证。

(Prop) Every closed set in ${\textstyle \mathbb {R} ^{n}}$ can be written as a countable union of compact sets.

Given a set ${\textstyle X}$, a collection ${\textstyle A}$ of subsets of ${\textstyle X}$ is called a ${\textstyle \sigma }$ algebra provided that

1. it contains the entire set and the empty set
2. closed under complement
3. closed under countable union

(Defn) Borel set

The collection ${\textstyle B}$ of Borel sets of real numbers is the smallest ${\textstyle \sigma }$ algebra of sets of real numbers that contains all of the open sets of real numbers.

Egoroff's theorem).