Countable Random Sets: Uniqueness in Law and Constructiveness

TL;DR

This paper investigates the uniqueness in law of countable random sets with accumulation points, comparing approaches via generators and hitting functions, and introduces constructiveness concepts including measurable selection and decomposition theorems.

Contribution

It provides new theorems on uniqueness in law for -finite and constructive countable random sets, including measurable selection and decomposition results.

Findings

Proved a measurable selection theorem for constructive countable random sets.

Established a decomposition theorem for constructive countable random sets.

Analyzed constructive countable random sets with independent increments.

Abstract

The first part of this article deals with theorems on uniqueness in law for \sigma-finite and constructive countable random sets, which in contrast to the usual assumptions may have points of accumulation. We discuss and compare two approaches on uniqueness theorems: First, the study of generators for \sigma-fields used in this context and, secondly, the analysis of hitting functions. The last section of this paper deals with the notion of constructiveness. We will prove a measurable selection theorem and a decomposition theorem for constructive countable random sets, and study constructive countable random sets with independent increments.