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 ofof subsets of
is called an algebra on
(or algebra of subsets of
) if
(i)
(ii),
(iii).
[Note thatand
]
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 collectionof subsets of
is called a
-algebra on
(or
-algebra of subsets of
) if
is an algebra on
such that whenever
, then
.
[Note that ifis 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
Letbe 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.