Boris Tsirelson,
"Filtrations of random processes in the light of classification theory. I. A topological zero-one law."
math.PR/0107121.
Available online (free of charge) from e-print archive (USA):
arXiv.org/abs/math.PR/0107121/
or its Israeli mirror:
il.arXiv.org/abs/math.PR/0107121/

A research preprint, 35 pages, bibl. 15 refs.

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.

1. Preliminaries on the classification theory (for probabilists).
2. Preliminaries on filtrations (for specialists in the classification theory).
3. Isomorphism of probability spaces as an orbit equivalence relation.
4. Isomorphism of filtrations as an orbit equivalence relation.