# σ-加法族とボレル集合体の定義を調べる

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．

-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.

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