  The recent extensions of domain theory have proved particularly efficient to
study lattice-valued maxitive measures, when the target lattice is continuous.
Maxitive measures are defined analogously to classical measures with the
supremum operation in place of the addition. Building further on the links
between domain theory and idempotent analysis highlighted by Lawson (2004), we
investigate the concept of domain-valued linear forms on an idempotent
(semi)module. In addition to proving representation theorems for continuous
linear forms, we address two applications: the idempotent Radon--Nikodym
theorem and the idempotent Riesz representation theorem. To unify similar
results from different mathematical areas, our analysis is carried out in the
general Z framework of domain theory.
