  In the context of averaging an inhomogeneous cosmological model, we propose a
natural measure identical to the Kullback-Leibler relative information entropy,
which expresses the distinguishability of the local inhomogeneous density field
from its spatial average on arbitrary compact domains. This measure is expected
to be an increasing function in time and thus to play a significant role in
studying gravitational entropy. To verify this conjecture, we explore the time
evolution of the measure using the linear perturbation theory of a spatially
flat FLRW model and a spherically symmetric nonlinear solution. We discuss the
generality and conditions for the time-increasing nature of the measure, and
also the connection to the backreaction effect caused by inhomogeneities.
