Spaces and maps of idempotent measures

We prove that the weak* topologization of the set of all idempotent measures (Maslov measures) on compact Hausdorff spaces defines a functor on the category $\operatorname{\mathbf{Comp}}$ of compact Hausdorff spaces, and this functor is normal in the sense of E. V. Shchepin; in particular, it has many properties in common with the probability measure functor and the hyperspace functor. Moreover, we establish that this functor defines a monad in the category $\operatorname{\mathbf{Comp}}$, and prove that the idempotent measure monad contains the hyperspace monad as a submonad. For the space of idempotent measures there is an analogue of the Milyutin map (that is, of a continuous map of compact Hausdorff spaces which admits a regular averaging operator for spaces of continuous functions). Using the assertion of the existence of Milyutin maps for idempotent measures, we prove that the idempotent measure functor is open, that is, it preserves the class of open surjective maps. We also prove that, in contrast to the case of probability measure spaces, the correspondence assigning to any pair of idempotent measures the set of measures on their product which have the given marginals is not continuous.