Why is the Bernoulli shift ergodic? I know the proof for $S^\mathbb{N}$. The best would be if you could argue from this fact to the integer case, so that way I can see how one usually goes from one-sided infinite to two-sided. (This seems to be coming up a good deal in my probability class.)
As a reminder, the Bernoulli shift is the left shift, regardless of if it's the natural numbers, where the first coordinate must be dropped, or the integers. Ergodicity is defined by all invariant sets having measure 0 or 1.
Any type of solution would be appreciated, even if it does not expand upon the natural number solution using the Kolmogorov 0-1 law on tail events. In particular, I have verified theorem 3 here, but cannot figure out how to check exercise 6. (I have the hint, but not the fact that the hint implies the claim in general.) http://terrytao.wordpress.com/2008/02/04/254a-lecture-9-ergodicity/