一〇二上實分析筆記@S-2013.09.23


 * 週三作業交到 1. 7. 2
 * Chapter 2 的作業為
 * 2. 1. 1,2
 * 2. 2. 2,5,6
 * 2. 3. 3
 * 2. 4. 1,2
 * 2. 5. 3,4,7,8,9
 * 2. 6. 2
 * 2. 7. 1,3,5,9,10
 * 上課會討論的習題為
 * 2. 2. 1,3,4
 * 2. 5. 2,5,6
 * 2. 6. 3
 * 2. 7. 2,4,6,11

§2.1
Family of sets and set functions. $\Omega$ is a universal set. $\Phi$ is collection of subsets of $\Omega$. $\tau:\Phi\to\mathbb R_{0+}=[0,\infty]$. $\tau$ is called a set function if $\tau(\varnothing)=0$. $\tau$ is monotonic if $\tau(A)\geq\tau(B)$ whenever $A\supset B$. A Monotonic set function is contonus at $A$ if for all $A_1\subset\dotsb A_n\subset\dotsb\subset A$ such that $\bigcup A_n=A$, $$\tau(A)=\lim\tau(A_n)$$

$\mathscr P\subset2^\Omega$ is called a $\Pi$-System (on $\Omega$) if $A\cap B\in\mathscr P$ whenever $A,B\in\mathscr P$.

For so called $\lambda$-system, $\mathscr P$ is closed under complement and disjoint union. For so called $\sigma$-system, $\mathscr P$ is closed under complement and union.

Remark
Since the intersection of two $X$-algebras is again an $X$-algebra, therefore there are always a smallest $X$-algebra on $\Omega$. Denoted them by $X(\Omega)$.

Lemma 2.1.1
A family is $\sigma$ if and only if it is $\pi$ and $\lambda$, if and only if it is $\lambda$ and is closed under $A\setminus B$.

Theorem 2.1.1
If $\mathscr P$ is $\pi$, then $\lambda(\mathscr P)=\sigma(\mathscr P)$.

§2.1 Measurable Space and Measurable Functions
Functions are defined on $\sigma$-algebra.