  We study a stability property of probability laws with respect to small
violations of algorithmic randomness. A sufficient condition of stability is
presented in terms of Schnorr tests of algorithmic randomness. Most probability
laws, like the strong law of large numbers, the law of iterated logarithm, and
even Birkhoff's pointwise ergodic theorem for ergodic transformations, are
stable in this sense. Nevertheless, the phenomenon of instability occurs in
ergodic theory. Firstly, the stability property of the Birkhoff's ergodic
theorem is non-uniform. Moreover, a computable non-ergodic measure preserving
transformation can be constructed such that ergodic theorem is non-stable. We
also show that any universal data compression scheme is also non-stable with
respect to the class of all computable ergodic measures.
