The strong topology of $\omega$-plurisubharmonic functions

On $(X,\omega )$ compact K"ahler manifold, given a model type envelope $\psi \in PSH(X,\omega )$ (i.e. a singularity type) we prove that the Monge-Amp`ere operator is an homeomorphism between the set of $\psi$-relative finite energy potentials and the set of $\psi$-relative 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 singularities types, the sets $X_{\mathcal{A}},Y_{\mathcal{A}}$ given respectively as the union of all $\psi$-relative finite energy potentials and of all $\psi$-relative finite energy measures varying $\psi \in \bar{\mathcal{A}}$ have two natural strong topologies which extends the strong topologies on each component of the unions. We show that the Monge-Amp`ere operator produces an homeomorphism between $X_{\mathcal{A}}$ and $Y_{\mathcal{A}}$. As an application we also prove the strong stability of a sequence of solutions of prescribed complex Monge-Amp`ere equations when the measures have uniformly $L^p$-bounded densities for $p>1$ and the prescribed singularities are totally ordered.