# 加法族の定義(旧)

## 有限加法族の定義

* $S$ is a set.
* $O :=\{F|F\subset S \}$
$O$ is called 有限加法族 if it fills following conditions.
1. $F \in O \Rightarrow F^c \in O$
2. $F, G \in O \Rightarrow F \cup G \in O$

## 命題

### Statement

* $S \in O$
* $\emptyset \in O$
* $F, G \in O \Rightarrow F - G \in O$
* $F, G \in O \Rightarrow F \cap G \in O$

easy

## 完全加法族の定義

($\sigma$-加法族、$\sigma$-algebra、可算加法族もしくは単に加法族ともいう)
$O$ is called 完全加法族 if it fills following conditions.
1. $F \in O \Rightarrow F^c \in O$
2. $F_n \in O \quad (n \in \mathbb{N}) \Rightarrow \cup_{n \in \mathbb{N}} F_n \in O$

## 命題

### Statement

* $S \in O$
* $\emptyset \in O$
* $F, G \in O \Rightarrow F - G \in O$
* $F_n \in O \quad (n \in \mathbb{N}) \Rightarrow \cap_{n \in \mathbb{N}} F_n \in O$

### Proof

$F_n \in O \quad (n \in \mathbb{N}) \Rightarrow F_n^c \in O \quad (n \in \mathbb{N}) \qquad (\because \mathbb{rule.1})$
$F_n^c \in O \quad (n \in \mathbb{N}) \Rightarrow \cup_{n \in \mathbb{N}} F_n^c \in O \qquad (\because \mathbb{rule.2})$
$\cup_{n \in \mathbb{N}} F_n^c \in O \Rightarrow (\cup_{n \in \mathbb{N}} F_n^c)^c \in O \qquad (\because \mathbb{rule.1})$
$(\cup_{n \in \mathbb{N}} F_n^c)^c \in O \Rightarrow \cap_{n \in \mathbb{N}} F_n \in O \qquad (\because \mathbb{De\ Morgan's\ laws})$