  We associate to every entanglement measure a family of measures which depend
on a precision parameter, and which we call epsilon-measures of entanglement.
Their definition aims at addressing a realistic scenario in which we need to
estimate the amount of entanglement in a state that is only partially known. We
show that many properties of the original measure are inherited by the family,
in particular weak monotonicity under transformations applied by means of Local
Operations and Classical Communication (LOCC). On the other hand, they may
increase on average under stochastic LOCC. Remarkably, the epsilon-version of a
convex entanglement measure is continuous even if the original entanglement
measure is not, so that the epsilon-version of an entanglement measure may be
actually considered a smoothed version of it.
