  Gudder, in a recent paper, defined a candidate entanglement measure which is
called the entanglement number. The entanglement number is first defined on
pure states and then it extends to mixed states by the convex roof
construction. In Gudder's article it was left as an open problem to show that
Optimal Pure State Ensembles (OPSE) exist for the convex roof extension of the
entanglement number from pure to mixed states. We answer Gudder's question in
the affirmative, and therefore we obtain that the entanglement number vanishes
only on the separable states. More generally we show that OPSE exist for the
convex roof extension of any function that is norm continuous on the pure
states of a finite dimensional Hilbert space. Further we prove that the
entanglement number is an LOCC monotone, (and thus an entanglement measure), by
using a criterion that was developed by Vidal in 2000. We present a simplified
proof of Vidal's result where moreover we use an interesting point of view of
tree representations for LOCC communications. Lastly, we generalize Gudder's
entanglement number by producing a monotonic family of entanglement measures
which converge in a natural way to the entropy of entanglement.
