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.



Measures that satisfy a 0/1 law
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.


