A new definition of random sets

TL;DR

This paper introduces a novel definition of random sets utilizing negative definite kernels and measurable space distances, diverging from traditional Hausdorff distance methods, to enhance the mathematical framework of random set theory.

Contribution

It proposes a new approach to defining random sets based on measurable space distances and negative definite kernels, offering an alternative to Hausdorff distance-based definitions.

Findings

Provides a new mathematical framework for random sets.

Establishes connections with Hilbert space isometries.

Offers potential for improved analysis of random sets.

Abstract

A new definition of random sets is proposed. It is based on the distance in measurable space and uses negative definite kernels for continuation from initial space to that of random sets. This approach has no connection to Hausdorff distance between sets. Key words: random sets; measurable space; negative definite kernels; Hilbert space isometries.