2001/07/17 by Boris Tsirelson, Tsirelson, Boris
Mathematics · #03E15 #60G07 #FOS: Mathematics #Logic (math.LO) #Probability (math.PR) #math.LO #math.PR #msc:03E15 #msc:60G07
paper · pdf · doi:10.48550/arxiv.math/0107121
35 pages
arxiv created 2001/07/17 · arxiv updated 2016/09/07
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.