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
$$ OH_+(X)=\Big\{\mu\in OH(X): \big\vert\mu(\varphi) \big\vert\le\mu\big(|\varphi| \big), \varphi\in C(X) \Big\}. $$
Under natural assumptions on $X$, we show that the triple
$$ \big(\mathcal{F}^\omega(X),\mathcal{F}^{++}(X),\mathcal{F}^+(X) \big) $$
is homeomorphic to $(Q,s,\mathrm{rint}\, Q)$, where $\mathcal{F}$ is a convex seminormal semimonadic subfunctor of $OH_+$.