  We characterize the points that satisfy Birkhoff's ergodic theorem under
certain computability conditions in terms of algorithmic randomness. First, we
use the method of cutting and stacking to show that if an element x of the
Cantor space is not Martin-Lof random, there is a computable measure-preserving
transformation and a computable set that witness that x is not typical with
respect to the ergodic theorem, which gives us the converse of a theorem by
V'yugin. We further show that if x is weakly 2-random, then it satisfies the
ergodic theorem for all computable measure-preserving transformations and all
lower semi-computable functions.
