Summary: On a compact Kähler manifold \((X,\omega)\), given a model-type envelope \(\psi\in \operatorname{PSH}(X,\omega)\) (i.e., a singularity type) we prove that the Monge-Ampère operator is a homeomorphism between the set of \(\psi \)-relative finite energy potentials and the set of \(\psi \)-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 \(\mathcal{A}\) of model-type envelopes with positive total mass representing different singularity types, the sets \(X_{\mathcal{A}}\) and \(Y_{\mathcal{A}} \), given as the union of all \(\psi \)-relative finite energy potentials and of all \(\psi \)-relative finite energy measures with varying \(\psi\in \bar{\mathcal{A}}\), 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 \(X_{\mathcal{A}}\) and \(Y_{\mathcal{A}} \). As an application we also prove the strong stability of a sequence of solutions of complex Monge-Ampère equations when the measures have uniformly \(L^p\)-bounded densities for \(p>1\) and the prescribed singularities are totally ordered.

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