A filtration is an indexed family of nested mathematical objects. In probability theory, a filtration 
on a probability space 
is a nondecreasing family of sigma-algebras
satisfying
 |
(1)
|
The sigma-algebra 
represents the information available by time 
. This form of filtration is used in the definitions of a stopping time, martingale,
and stochastic integrals.
In algebra, a filtration of ideals of a commutative ring 
with multiplicative
identity is a sequence of ideals
 |
(2)
|
such that 
for all indices
. An example is the 
-adic filtration associated with a proper
ideal 
of 
,
 |
(3)
|
A ring equipped with a filtration is called a filtered ring.
