Borel-Cantelli Lemma: Difference between revisions

From Optimal Transport Wiki
Jump to navigation Jump to search
(Created page with "=== '''ICTP Real Analysis''' === <blockquote>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...")
 
(Blanked the page)
Tag: Blanking
Line 1: Line 1:
=== '''ICTP Real Analysis''' ===


<blockquote>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。
</blockquote>
__TOC__
==== '''Part 0''' ====
<blockquote>sigma-algebra <math display="inline">\to</math> Borel set,
<math display="inline">G_\delta</math> set, <math display="inline">F_\sigma</math> set.
</blockquote>
(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<math display="inline">x</math>，定义 <math display="inline">a_x := \inf\{z|(z,x)\subseteq O\}</math> 和 <math display="inline">a_x := \sup\{y|(x,y)\subseteq O\}</math> 且<math display="inline">I_x:=(a_x,b_x)</math>. 易证<math display="inline">a_x, b_x \notin O</math>. 根据有理数在实数中的稠密性，可在 <math display="inline">I_x</math> 和有理数<math display="inline">k\in I_x</math> 之间建立一一对应关系。显然<math display="inline">\{I_x\}</math> '''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 <math display="inline">\R^n</math> can be written as a countable union of compact sets.
Given a set <math display="inline">X</math>, a collection <math display="inline">A</math> of subsets of <math display="inline">X</math> is called a <math display="inline">\sigma</math> algebra provided that
# it contains the entire set and the empty set
# closed under complement
# closed under countable union
(Defn) Borel set
The collection <math display="inline">B</math> of Borel sets of real numbers is the smallest <math display="inline">\sigma</math> algebra of sets of real numbers that contains all of the open sets of real numbers.
Egoroff's theorem).

Revision as of 06:00, 17 December 2020

Retrieved from "http://www.otwiki.xyz/index.php?title=Borel-Cantelli_Lemma&oldid=1751"