  We develop two notions of time-restricted sensitivity to initial conditions
for measurable dynamical systems, where the time before divergence of a pair of
paths is at most an asymptotically logarithmic function of a measure of their
initial distance. In the context of finite measure-preserving transformations
on a compact space, we relate these notions to the metric entropy of the
system. We examine one of these notions for classes of non-measure-preserving,
nonsingular transformations.
