  This paper studies the notion of W-measurable sensitivity in the context of
semigroup actions. W-measurable sensitivity is a measurable generalization of
sensitive dependence on initial conditions. In 2012, Grigoriev et. al. proved a
classification result of conservative ergodic dynamical systems that states all
are either W-measurably sensitive or act by isometries with respect to some
metric and have refined structure. We generalize this result to a class of
semigroup actions. Furthermore, a counterexample is provided that shows
W-measurable sensitivity is not preserved under factors. We also consider the
restriction of W-measurably sensitive semigroup actions to sub-semigroups and
show that the restriction remains W-measurably sensitive when the sub-semigroup
is large enough (e.g. when the sub-semigroups are syndetic or thick).
