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₊.

中文翻译：

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正齐次泛函的函子的三重无限迭代

本文致力于度量公制紧致矩阵X上所有弱加阶保持标准化正均匀功能的空间OH（X）的研究。我们通过Kantorovich-Rubinshteĭn度量证明了函子OH的均匀可度量性。我们还证明了函子OH ₊是完全可度量的，其中$$ O {H_ +} \ left（X \ right）= \ left \ {{\ mu \; \ in \; OH \ left（X \ right）\;：\; \ left | {\ mu \ left（\ varphi \ right）} \ right | \; \ le \; \ mu \ left（{\ left | \ varphi \ right |} \ right），\; \ varphi \; \ in \; C \ left（X \ right）} \ right \}。$$在X的自然假设下，我们表明三元组$$ \ left（{{{\ cal F} ^ \ omega} \ left（X \ right），\; {{\ cal F} ^ {+ +}} \ left（X \ right），\; {{ \ cal F} ^ +} \ left（X \ right）} \ right）$$是（Q，s，rint Q）的同胚，其中\（{\ cal F} \）是OH的凸半正半单子函数₊。