  On spaces of finite signed Borel measures on a metric space one has
introduced the Fortet-Mourier and Dudley norms, by embedding the measures into
the dual space of the Banach space of bounded Lipschitz functions, equipped
with different -- but equivalent -- norms: the FM-norm and the BL-nor,
respectively. The norm of such a measure is then obtained by maximising the
value of the measure when applied by integration to extremal functions of the
unit ball. We introduce suitable subsets of extremal functions in both cases,
essentially by a construction by means of McShane's Lipschitz extension
operator for Lipschitz functions on finite metric spaces, that are shown to be
norming. As a consequence, we obtain that these sets are weak-star dense in the
full set of extreme points of the unit ball. If the underlying metric space is
connected, then we have found a larger set of extremal functions for the
BL-norm, similar to such a set that was defined previously by J. Johnson for
the FM-norm when the underlying space was compact. This set is then also
weak-star dense in the extremal functions. This may open an avenue to obtaining
computational approaches for the Dudley norm on signed Borel measures.
