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(-algebra)とボレル集合体(Borel -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 be a set.
Algebra on
A collection of of subsets of is called an algebra on (or algebra of subsets of ) if
(i)
(ii) ,
(iii) .
[Note that and
]
ちなみに,以上より
が得られ,さらにこれを用いて
が得られます.
先に進みます. です.
-algebra on
A collection of subsets of is called a -algebra on (or -algebra of subsets of ) if is an algebra on such that whenever , then.
[Note that if is a -algebra on and for , then
.]
Thus, a -algebra on is a family of subsets of stable under any countable collection of set operations.
, -algebra generated by a class of subsets
Let be a class of subsets of . Then , the -algebra generated by , is the smallest -algebra on such that . It is the intersection of all -algebras on which have as a subclass.(Obviously, the class of all subsets of is a -algebra which extends .)
Let be a topological space.
, the Borel -algebra on , is the -algebra generated by the familiy of open subsets of . With slight abuse of notation,
:= open sets.
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.