  An outlier is a datapoint that is set apart from a sample population. The
outlier theorem in algorithmic information theory states that given a
computable sampling method, outliers must appear. We present a simple proof to
the outlier theorem, with exponentially improved bounds. We extend the outlier
theorem to ergodic dynamical systems which are guaranteed to hit ever larger
outlier states with diminishing measures. We show how to construct
deterministic functions from random ones, i.e. function derandomization. We
also prove that all open sets of the Cantor space with large uniform measure
will either have a simple computable member or high mutual information with the
halting sequence.
