A theorem of Ku\v{c}era states that given a Martin-L\"of random infinite
binary sequence {\omega} and an effectively open set A of measure less than 1,
some tail of {\omega} is not in A. We first prove several results in the same
spirit and generalize them via an effective version of a weak form of
Birkhoff's ergodic theorem. We then use this result to get a stronger form of
it, namely a very general effective version of Birkhoff's ergodic theorem,
which improves all the results previously obtained in this direction, in
particular those of V'Yugin, Nandakumar and Hoyrup, Rojas.