  The Reduction Map Theorem in H. Tsuji's work on numerical trivial fibrations
is corrected and proven. To this purpose various definitions of Tsuji's new
intersection numbers for pseudo-effective line bundles equipped with a positive
singular hermitian metric are compared and their equivalence on sufficiently
general smooth curves is shown. Numerically trivial varieties are characterized
by a decomposition property of the curvature current. An important adjustment
to the Reduction Map Theorem is to consider the fact that plurisubharmonic
functions are singular on pluripolar sets.
