Measures that satisfy a 0/1 law

Question

The setting is measure on $2^\omega$. That product (independent) measures obey a 0/1 law, i.e, that measurable tail sets all have measure 0 or 1, is well known. I've made some progress extending this to measures that satisfy a weak symmetry property that's a little complicated to state, but roughly is that the limit of the ratio of measures of finite initial segments, along two infinite binary sequences that are eventually equal, doesn't get too big or too small.

At any rate, It would be helpful to know how much progress has been made extending 0/1 laws to non-independent atomless measures. I've researched this a bit and haven't found anything, but of course that doesn't mean it hasn't been done.

Answer 1

These measures are well studied in ergodic theory. They are measures with the $K$ (for Kolmogorov) property. It's known that they are a bit of a zoo from an ergodic point of view: Ornstein's celebrated theorem for Bernoulli shifts says that two Bernoulli shifts are isomorphic as measure-preserving systems if and only if they have the same entropy, whereas it was proved soon afterwards that there are uncountably many $K$ systems with equal entropy.

There are several equivalent formulations of the K property. A good reference is Rudolph's book: Introduction to Measurable Dynamics.

Answer 2

I would like to point out that extremal Gibbs measures are interesting class of measures that can be defined over $2^{\omega}$ and satisfies a zero-one law, not in the whole $\sigma$-algebra generated by the cylinder sets, but in the tail $\sigma$-algebra.

In several cases this approach agree with the Anthony suggestion. In the Gibbs measure theory dependence is constructed in very geometric way. Another interesting feature is: no group action invariance is required. Some theorems of ergodic theory are proved in this context. There is also a Choquet's Theorem.

If you want to get the details the classical mathematical reference is Gibbs Measures and Phase Transitions by Hans-Otto Georgii. The zero-one law is proved in the page 115 of the first edition.

There are some online options which are not so general as Georgii, but covers the space you are interested in and proves the zero-one law

A. Bovier: Lecture notes Gibbs measures and phase transitions - part 1.
http://www-wt.iam.uni-bonn.de/~bovier/files/note1.pdf

A. Bovier: Lectures notes Gibbs measures and phase transitions - part 2.
http://www-wt.iam.uni-bonn.de/~bovier/files/note2.pdf