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

easy

Definition 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

* $F \in O \Rightarrow F^c \in O$
* $\emptyset \in O$
* $F_n \in O \quad (n \in \mathbb{N}) \Rightarrow \cap_{n \in \mathbb{N}} F_n \in O$

Proof

easy