  Given an iterated function system (IFS) on a complete and separable metric
space $Y$, there exists a unique compact subset $X \subseteq Y$ satisfying a
fixed point relation with respect to the IFS. This subset is called the
attractor set, or fractal set, associated to the IFS. The attractor set
supports a specific Borel probability measure, called the Hutchinson measure,
which itself satisfies a fixed point relation. P. Jorgensen generalized the
Hutchinson measure to a projection-valued measure, under the assumption that
the IFS does not have essential overlap. In previous work, we developed an
alternative approach to proving the existence of this projection-valued
measure. The situation when the IFS exhibits essential overlap has been studied
by Jorgensen and colleagues in. We build off their work to generalize the
Hutchinson measure to a positive-operator valued measure for an arbitrary IFS,
that may exhibit essential overlap. This work hinges on using a generalized
Kantorovich metric to define a distance between positive operator-valued
measures. It is noteworthy to mention that this generalized metric, which we
use in our previous work as well, was also introduced by R.F. Werner to study
the position and momentum observables, which are central objects of study in
the area of quantum theory. We conclude with a discussion of Naimark's dilation
theorem with respect to this positive operator-valued measure, and at the
beginning of the paper, we prove a metric space completion result regarding the
classical Kantorovich metric.

Functional Analysis