  We define a (mean curvature flow) entropy for Radon measures in
$\mathbb{R}^n$ or in a compact manifold. Moreover, we prove a monotonicity
formula of the entropy of the measures associated with the parabolic Allen-Cahn
equations. If the ambient manifold is a compact manifold with non-negative
sectional curvature and parallel Ricci curvature, this is a consequence of a
new monotonicity formula for the parabolic Allen-Cahn equation. As an
application, we show that when the entropy of the initial data is small enough
(less than twice of the energy of the one-dimensional standing wave), the limit
measure of the parabolic Allen-Cahn equation has unit density for all future
time.
