I changed my question with more details.
I think to show this collection is a field, I have to show the complements of each elements is again in this set and the union of each element is also in this set. But I think this properties are for sigma field. Are they valid also for field ?
Or to...
1. Show that Ao, the collection of all cylinders of all rank is a field.
A cylinder of rank n is a set of the form { w∈S^∞ : R1(w)R2(w)...Rn(w) ∈ H}
where H is a set of n-long sequences of elements of S. That is H is a subset of S^n
Example:
now think about a toss a coin question...