=+15. Let Y1, Y2,... be a sequence of independent random variables. The tail -algebra T generated by

=+15. Let Y1, Y2,... be a sequence of independent random variables. The tail

σ-algebra T generated by the sequence can be expressed as T = ∩nTn, where Tn is the σ-algebra generated by Yn, Yn+1,.... It is easy to construct events in T . For instance in an infinite sequence of coin tosses, the event that a finite number of heads occurs belongs to T . The zero-one law says that any C ∈ T has Pr(C) = 0 or Pr(C) = 1.

Use Proposition 10.3.2 and Problem 14 to prove the zero-one law.

(Hints: Let Fn be the σ-algebra generated by Y1,...,Yn. The martingale Xn = E(1C | Fn) is constant and converges to X∞, which is measurable with respect to F∞. But C is a member of F∞.)