Let S = R[0,1], the set of functions from [0, 1] to R, and let F be the σ-field generated by the cylindrical sets. The purpose of this exercise is to show that the elements of F depend on only...

Let S = R[0,1], the set of functions from [0, 1] to R, and let F be the σ-field generated by the cylindrical sets. The purpose of this exercise is to show that the elements of F depend on only countably many coordinates.

Let
  the set of sequences taking values in R. Let F₀
be the σ-field generated by the cylindrical subsets of R^N, where N = {1, 2,...}.

Show that B ∈ F if and only if there exist t₁,t₂,... in
  and a set
 such that