A Triple of Infinite Iterates of the Functor of Positively Homogeneous Functionals

The present article is devoted to the study of the space OH(X) of all weakly additive order-preserving normalized positively homogeneous functionals on a metric compactum X. We prove the uniform metrizability of the functor OH by means of the Kantorovich-Rubinshteĭn metric. We also show that the functor OH+ is perfectly metrizable, where $$O{H_ +}\left(X \right) = \left\{{\mu \; \in \;OH\left(X \right)\;:\;\left| {\mu \left(\varphi \right)} \right|\; \le \;\mu \left({\left| \varphi \right|} \right),\;\varphi \; \in \;C\left(X \right)} \right\}.$$ Under natural assumptions on X, we show that the triple $$\left({{{\cal F}^\omega}\left(X \right),\;{{\cal F}^{+ +}}\left(X \right),\;{{\cal F}^ +}\left(X \right)} \right)$$ is homeomorphic to (Q, s, rint Q), where $${\cal F}$$ is a convex seminormal semimonadic subfunctor of OH+.

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