読者です 読者をやめる 読者になる 読者になる

エンジニアを目指す浪人のブログ

情報系に役立ちそうな応用数理をゆるめにメモします

σ-加法族とボレル集合体の定義を調べる

測度論や確率論における重要な概念としてσ-加法族({\displaystyle \sigma}-algebra)とボレル集合体(Borel {\displaystyle \sigma}-algebra)があります.それらについて頭を整理するべく調べていたのですが,Williams(1991)の記述が明快ですので,少し長いですが引用します.

Let {\displaystyle \mathcal{S}} be a set.
Algebra on {\displaystyle \mathcal{S}}
A collection of {\displaystyle \Sigma_0} of subsets of {\displaystyle \mathcal{S}} is called an algebra on {\displaystyle \mathcal{S}} (or algebra of subsets of {\displaystyle \mathcal{S}}) if
  (i) {\displaystyle \mathcal{S} \in \Sigma_0}
 (ii) {\displaystyle F \in \Sigma_0 \;\; \Rightarrow \;\; F^c := \mathcal{S} \setminus F \in  \Sigma_0},
(iii) {\displaystyle F,G \in \Sigma_0  \;\; \Rightarrow \;\; F \cup G \in \Sigma_0}.
[Note that {\displaystyle \emptyset = \mathcal{S}^c \in \Sigma_0} and
{\displaystyle F,G \in \Sigma_0 \;\; \Rightarrow \;\; F \cap G = (F^c \cup G^c)^c \in  \Sigma_0 .}

ちなみに,以上より
          {\displaystyle F,G \in \Sigma_0  \;\; \Rightarrow \;\; F \setminus G = F \cap G^c \in \Sigma_0 }
が得られ,さらにこれを用いて
          {\displaystyle F,G \in \Sigma_0  \;\; \Rightarrow \;\; F \bigtriangleup G = (F \setminus G) \cup (G \setminus F) \in \Sigma_0 }
が得られます.
先に進みます.{\displaystyle \mathsf{N} := \{ 1,2,\cdots  \} } です.

 {\displaystyle \sigma}-algebra on {\displaystyle \mathcal{S}}
A collection {\displaystyle \Sigma} of subsets of {\displaystyle \mathcal{S}} is called a {\displaystyle \sigma}-algebra on {\displaystyle \mathcal{S}} (or {\displaystyle \sigma}-algebra of subsets of {\displaystyle \mathcal{S}}) if {\displaystyle \Sigma} is an algebra on {\displaystyle \mathcal{S}} such that whenever {\displaystyle F_n \in \Sigma \; (n \in \mathsf{N}) }, then

{\displaystyle \bigcup_n F_n \in \Sigma}.
[Note that if {\displaystyle \Sigma} is a {\displaystyle \sigma}-algebra on {\displaystyle \mathcal{S}} and {\displaystyle F_n \in \Sigma}  for  {\displaystyle n \in \mathsf{N}}, then
{\displaystyle \bigcap_n F_n = ( \bigcup_n F_n^c )^c \in \Sigma} .]
Thus, a {\displaystyle \sigma}-algebra on {\displaystyle \mathcal{S}} is a family of subsets of {\displaystyle \mathcal{S}} stable under any countable collection of set operations.

{\displaystyle \sigma(\mathcal{C})}, {\displaystyle \sigma}-algebra generated by a class {\displaystyle \mathcal{C}} of subsets
Let {\displaystyle \mathcal{C}} be a class of subsets of {\displaystyle \mathcal{S}}. Then {\displaystyle \sigma(\mathcal{C})}, the {\displaystyle \sigma}-algebra generated by {\displaystyle \mathcal{C}}, is the smallest {\displaystyle \sigma}-algebra {\displaystyle \Sigma} on {\displaystyle \mathcal{S}} such that {\displaystyle \mathcal{C} \subseteq \Sigma}. It is the intersection of all {\displaystyle \sigma}-algebras on {\displaystyle \mathcal{S}} which have {\displaystyle \mathcal{C}} as a subclass.(Obviously, the class of all subsets of {\displaystyle \mathcal{S}} is a {\displaystyle \sigma}-algebra which extends {\displaystyle \mathcal{C}}.) 

Let {\displaystyle \mathcal{S}} be a topological space.
{\displaystyle \mathcal{B}(\mathcal{S})}
{\displaystyle \mathcal{B}(\mathcal{S})}, the Borel {\displaystyle \sigma}-algebra on {\displaystyle \mathcal{S}}, is the {\displaystyle \sigma}-algebra generated by the familiy of open subsets of {\displaystyle \mathcal{S}}. With slight abuse of notation,
{\displaystyle \mathcal{B}(\mathcal{S})} := {\displaystyle \sigma(}open sets{\displaystyle )}.

 

注意点は,σ-加法族は一般の集合に対する概念ですが,ボレルσ-加法族(ボレル集合体)は位相空間に対する概念である(開集合から生成されている!)ことでしょうか.測度論の立場からはより厳密な議論が必要でしょうが,応用上はこれらの定義を抑えておけば問題ない場合も多いと思います.

 

参考文献
[1] Williams, D. (1991), Probability with Martingales, Cambridge University Press.