  The set of all idempotent probability measures (Maslov measures) on a compact
Hausdorff space endowed with the weak* topology determines is functorial on the
category $\comp$ of compact Hausdorff spaces. We prove that the obtained
functor is normal in the sense of E. Shchepin. Also, this functor is the
functorial part of a monad on $\comp$. We prove that the idempotent probability
measure monad contains the hyperspace monad as its submonad. A counterpart of
the notion of Milyutin map is defined for the idempotent probability measures.
Using the fact of existence of Milyutin maps we prove that the functor of
idempotent probability measures preserves the class of open surjective maps.
Unlikely to the case of probability measures, the correspondence assigning to
every pair of idempotent probability measures on the factors the set of
measures on the product with these marginals, is not open.
