The structures of Hausdorff metric in non-Archimedean spaces

TL;DR

This paper explores the Hausdorff metric structures in non-Archimedean spaces, constructing new metrics and analyzing their properties, including convergence and algebraic structures, with applications to measure spaces.

Contribution

It introduces several novel metric structures on spaces related to non-Archimedean spaces and studies their properties and relations, including a Dudly type metric for measures.

Findings

Constructed new metric families like \hat{\rho}_u and \hat{\beta}_{X,Y}^\lambda.

Analyzed convergence and structural relations of these metrics.

Developed a Dudly type metric for K-valued measures on compact non-Archimedean spaces.

Abstract

For non-Archimedean spaces $ X $ and $ Y, $ let $ \mathcal{M}_{\flat } (X), \mathfrak{M}(V \rightarrow W) $ and $ \mathfrak{D}_{\flat }(X, Y) $ be the ballean of $ X $ (the family of the balls in $ X $), the space of mappings from $ X $ to $ Y, $ and the space of mappings from the ballen of $ X $ to $ Y, $ respectively. By studying explicitly the Hausdorff metric structures related to these spaces, we construct several families of new metric structures (e.g., $ \widehat{\rho } _{u}, \widehat{\beta }_{X, Y}^{\lambda }, \widehat{\beta }_{X, Y}^{\ast \lambda } $) on the corresponding spaces, and study their convergence, structural relation, law of variation in the variable $ \lambda, $ including some normed algebra structure. To some extent, the class $ \widehat{\beta }_{X, Y}^{\lambda } $ is a counterpart of the usual Levy-Prohorov metric in the probability measure spaces, but it behaves very differently, and is interesting in itself. Moreover, when $ X $ is compact and $ Y = K $ is a complete non-Archimedean field, we construct and study a Dudly type metric of the space of $ K-$valued measures on $ X. $