  We initiate the study of effective pointwise ergodic theorems in
resource-bounded settings. Classically, the convergence of the ergodic averages
for integrable functions can be arbitrarily slow. In contrast, we show that for
a class of PSPACE L1 functions, and a class of PSPACE computable
measure-preserving ergodic transformations, the ergodic average exists for all
PSPACE randoms and is equal to the space average on every EXP random. We
establish a partial converse that PSPACE non-randomness can be characterized as
non-convergence of ergodic averages. Further, we prove that there is a class of
resource-bounded randoms, viz. SUBEXP-space randoms, on which the corresponding
ergodic theorem has an exact converse - a point x is SUBEXP-space random if and
only if the corresponding effective ergodic theorem holds for x.
