  Although a number of measures for quantum non-Markovianity have been proposed
recently, it is still an open question whether these measures directly
characterize the memory effect of the environment, i.e., the dependence of a
quantum state on its past in a time evolution. In this paper, we present a
criterion and propose a measure for non-Markovianity with clear physical
interpretations of the memory effect. The non-Markovianity is defined by the
inequality $T(t_2,t_0)\neq T(t_2,t_1)T(t_1,t_0)$ in terms of memoryless
dynamical map $T$ introduced in this paper. This definition is conceptually
distinct from that based on divisibility used by Rivas et al (Phys. Rev. Lett
105, 050403 (2010)), whose violation is manifested by non-complete positivity
of the dynamical map. We demonstrate via a typical quantum process that without
Markovian approximation, nonzero memory effects (non-Markovianity) always exist
even if the non-Markovianity is zero by the other non-Marovianity measures.
