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 igtriangleup 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 igcup_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 igcap_n F_n = ( igcup_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.