  Although many different entanglement measures have been proposed so far, much
less is known in the multipartite case, which leads to the previous monogamy
relations in literatures are not complete. We establish here a strict framework
for defining multipartite entanglement measure (MEM): apart from the postulates
of bipartite measure, a genuine MEM should additionally satisfy the unification
condition and the hierarchy condition. We then come up with a complete monogamy
formula for the unified MEM and a tightly complete monogamy relation for the
genuine MEM. Consequently, we propose MEMs which are multipartite extensions of
entanglement of formation (EoF), concurrence, tangle, Tsallis $q$-entropy of
entanglement, R\'{e}nyi $\alpha$-entropy of entanglement, the convex-roof
extension of negativity and negativity, respectively. We show that (i) the
extensions of EoF, concurrence, tangle, and Tsallis $q$-entropy of entanglement
are genuine MEMs, (ii) multipartite extensions of R\'{e}nyi $\alpha$-entropy of
entanglement, negativity and the convex-roof extension of negativity are
unified MEMs but not genuine MEMs, and (iii) all these multipartite extensions
are completely monogamous and the ones which are defined by the convex-roof
structure (except for the R\'{e}nyi $\alpha$-entropy of entanglement and the
convex-roof extension of negativity) are not only completely monogamous but
also tightly completely monogamous. In addition, we find a class of tripartite
states that one part can maximally entangled with other two parts
simultaneously according to the definition of maximally entangled mixed state
(MEMS) in [Quantum Inf. Comput. 12, 0063 (2012)]. Consequently, we improve the
definition of maximally entangled state (MES) and prove that there is no MEMS
and that the only MES is the pure MES.
