# フィルトレーションの定義を調べるその1(離散時間)

Williams(1991)から引用します． です．

10.1 Filtered spaces
As basic datum, we now take a filtered space . Here,
is a probability triple as usual,
is a filtration, that is, an increasingly family of sub--algebras of :
.
We define
.

Intuitive idea. The information about in available to us at (or, if you prefer, just after) time consists precisely of the values of for all measurable function . Usuallly, is the natural filtration

of some (stochastic) process , and then the information about which we have at time consists of the values

・フィルトレーション は  の部分-加法族であり，確率過程により生成される -加法族(wikipediaも参照ください)である
・ある時点において利用できる に関する情報は，その時点までの確率変数(の実現値の)列である

ついでに，適合過程の定義も載せておきます．

10.2 Adapted process
A process is called adapted (to the filtration ) if for each , is -measurable.
Intuitive idea. If is adapted, the value is known to us at time . Usually, and for some -measurable function on .