Measures of finite energy in pluripotential theory: a synthetic approach

We introduce a synthetic approach to global pluripotential theory, covering in particular the case of a compact Kähler manifold and that of a projective Berkovich space over a non-Archimedean field. We define and study the space of measures of finite energy, introduce twisted energy and free energy functionals thereon, and show that coercivity of these functionals is an open condition with respect to the polarization.