  Filtered probability spaces (called "filtrations" for short) are shown to
satisfy such a topological zero-one law: for every property of filtrations,
either the property holds for almost all filtrations, or its negation does. In
particular, almost all filtrations are conditionally nonatomic.
  An accurate formulation is given in terms of orbit equivalence relations on
Polish G-spaces. The set of all isomorphic classes of filtrations may be
identified with the orbit space X/G for a special Polish G-space X. A "property
of filtrations" means a G-invariant subset of X having the Baire property.
"Almost all filtrations" means a comeager subset of X (the Baire category
approach). The zero-one law is a kind of ergodicity of X. It holds for
filtrations both in discrete and continuous time.
  The interplay between probability theory and descriptive set theory could be
interesting for both parties.
