  Given a compact metric space $X$, the collection of Borel probability
measures on $X$ can be made into a compact metric space via the Kantorovich
metric. We partially generalize this well known result to projection-valued
measures. In particular, given a Hilbert space $\mathcal{H}$, we consider the
collection of projection-valued measures from $X$ into the projections on
$\mathcal{H}$. We show that this collection can be made into a complete and
bounded metric space via a generalized Kantorovich metric. However, we add that
this metric space is not compact, thereby identifying an important distinction
from the classical setting. We have seen recently that this generalized metric
has been previously defined by F. Werner in the setting of mathematical
physics. To our knowledge, we develop new properties and applications of this
metric. Indeed, we use the Contraction Mapping Theorem on this complete metric
space of projection-valued measures to provide an alternative method for
proving a fixed point result due to P. Jorgensen. This fixed point, which is a
projection-valued measure, arises from an iterated function system on $X$, and
is related to Cuntz Algebras.
