planemath上关于progressive stochastic process的解释:
A stochastic process 
is said to be adapted to a filtration 
on the measurable space 
if 
is an 
-measurable random variable for each 
. However, for continuous-time processes, where the time 
ranges over an arbitrary index set 
, the property of being adapted is too weak to be helpful in many situations. Instead, considering the process as a map
it is useful to consider the measurability of 
.
The process 
is progressive or progressively measurable if, for every 
, the stopped process 
is 
-measurable. In particular, every progressively measurable process will be adapted and jointly measurable. In discrete time, when 
is countable, the converse holds and every adapted process is progressive.
A set 
is said to be progressive if its characteristic function 
is progressive. Equivalently,
for every 
. The progressively measurable sets form a 
-algebra, and a stochastic process is progressive if and only if it is measurable with respect to this 
-algebra.
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![$\displaystyle S\cap\left( (-\infty,s]\times\Omega\right)\in\mathcal{B}(\mathbb{T})\otimes\mathcal{F}_s$](http://images.planetmath.org:8080/cache/objects/11362/l2h/img18.png "$\displaystyle S\cap\left( (-\infty,s]\times\Omega\right)\in\mathcal{B}(\mathbb{T})\otimes\mathcal{F}_s$"){kind=small}
