In particular, if the underlying probability space is finite (i.e. is finite), the minimal sets of (with respect to set inclusion) are given by the union over all of the sets of minimal sets of that lie in . It can be shown that is -measurable. However, simple examples show that, in general, . If and are stopping times on , and almost surely, then See also Natural filtration Filtration (probability theory) Filter (mathematics) References Algebra Measure theory Stochastic processes