A~triple of~infinite iterates of~the~functor of~positively homogeneous functionals

G. F. Djabbarov

Matematičeskie trudy, Tome 22 (2019) no. 1, pp. 101-118

Résumé

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 $OH_+$.