随机过程的过滤问题及其分类理论。I.拓扑零一律
Filtrations of random processes in the light of classification theory. I. A topological zero-one law
摘要：
过滤概率空间（简称“过滤”）满足这样一个拓扑零一律：对于过滤的每一个性质，要么该性质对几乎所有过滤成立，要么其否定成立。特别是，几乎所有的过滤都是条件非原子的。利用波兰G-空间上的轨道等价关系给出了一个精确的公式。对于一个特殊的Polish G-空间X，所有同构的滤子类的集合可以用轨道空间X/G来表示。“滤子的性质”是指X的G不变子集具有Baire性质。“几乎所有的过滤”意味着X的一个集合子集（Baire范畴方法）。零一律是X的一种遍历性。它适用于离散时间和连续时间的过滤。概率论和描述集合论之间的相互作用可能对双方都很有趣。
英文摘要：
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.
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