[Probability Theory1] Week1

서울대학교 통계학과 대학원 2021년도 1학년 1학기 확률론1 수업의 정리내용입니다.

reference: Lecture note (Prof.sangSangyeol Lee),
Probability: Theory and Examples, Rick Durrett, Version 5 January 11, 2019

sigma field

$\textbf{Definition}$

Let $\Omega$ be a space or set. A family of subsets of $\Omega$, named $\mathcal{F}$, is a $\sigma$-field if
$\ \ \ \text{1) } \ \Omega \in \mathcal{F} $
$\ \ \ \text{2) } \ A \in \mathcal{F} \rightarrow A^c \in \mathcal{F} $
$\ \ \ \text{3) } \ A_i \in \mathcal{F},\ \text{i =1, 2,} \cdots \rightarrow \cup ^\infty _{i=1} A_i = \mathcal{F}$

$\textbf{Definition}$

Let $\varphi$ be a subclass of subsets of $\Omega$. $\sigma(\varphi)$ is the smallest $\sigma$-field coniaining $\varphi$. that is, $\sigma(\varphi) = \cap_{\varphi \subset g : \sigma-field} g$

$\textbf{Definition}$

Let $\mathbb{R}$ be the set of real numbers. And let $\tau$ be a ‘topology’ of $\mathbb{R}$; that is, $\tau = \lbrace A \subset \mathbb{R} : A$ is open $\rbrace$. Then $\sigma(\varphi)$ is called Borel field

$\pi - \lambda~$ System Theorem

$\textbf{Definition[pi-system]}$

A collection $\mathscr{P}$ of subsets of $\Omega$ is a $\pi$-system if $\forall A,B \in \mathscr{P}, ~~ A\cap B \in \mathscr{P}$

$\textbf{Definition[lambda-system]}$

A collection $\mathscr{L}$ of subsets of $\Omega$ is a $\lambda$-system if

1. $\Omega \in \mathscr{L}$
2. $A \in \mathscr{L} \rightarrow A^c \in \mathscr{L}$
3. $A_i \in \mathscr{L}, i=1,2, \cdots , A_i \cap A_j = \emptyset, \rightarrow \cup^{\infty}_{i=1} A_i \in \mathscr{L}$