On a compact Kähler manifold
,
given a model-type envelope
(i.e., a singularity type) we prove that the Monge–Ampère operator is a homeomorphism between the set of
-relative finite energy potentials
and the set of
-relative
finite energy measures endowed with their strong topologies given as
the coarsest refinements of the weak topologies such that the relative
energies become continuous. Moreover, given a totally ordered family
of
model-type envelopes with positive total mass representing different singularity types, the
sets
and
, given as the union of all
-relative finite energy potentials
and of all
-relative finite
energy measures with varying
,
respectively, have two natural strong topologies which extend the
strong topologies on each component of the unions. We show that
the Monge–Ampère operator produces a homeomorphism between
and
.
As an application we also prove the strong stability of a sequence of solutions
of complex Monge–Ampère equations when the measures have uniformly
-bounded
densities for
and the prescribed singularities are totally ordered.
