加法族
Definition 有限加法族
* $S$ is a set.* $O :=\{F|F\subset S \}$
$O$ is called 有限加法族 if it fills following conditions.
1. $F, G \in O \Rightarrow F \cup G \in O$
2. $F, G \in O \Rightarrow F - G \in O$
3. $S \in O$
命題
Statement
* $F \in O \Rightarrow F^c \in O$* $\emptyset \in O$
* $F, G \in O \Rightarrow F \cap G \in O$
Proof
easyDefinition 完全加法族
($\sigma$-加法族、$\sigma$-algebra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. $F_n \in O \quad (n \in \mathbb{N}) \Rightarrow \cup_{n \in \mathbb{N}} F_n \in O$
2. $F, G \in O \Rightarrow F - G \in O$
3. $S \in O$
命題
Statement
* $F \in O \Rightarrow F^c \in O$* $\emptyset \in O$
* $F_n \in O \quad (n \in \mathbb{N}) \Rightarrow \cap_{n \in \mathbb{N}} F_n \in O$