A note on compact and {\sigma}-compact subsets of probability measures on metric spaces with an application to the distribution free newsvendor problem

TL;DR

This paper characterizes compact and {}-compact subsets of probability measures on metric spaces under weak convergence and applies these findings to extend the distribution free newsvendor problem.

Contribution

It provides new insights into the structure of probability measure spaces and demonstrates their application to a classic inventory management problem.

Findings

Characterization of compact and -compact subsets of probability measures

Example showing probability measure space on -compact metric spaces may not be -compact

Application to extended distribution free newsvendor problem

Abstract

This note identifies compact and {\sigma}-compact subsets of probability measures on a class of metric spaces with respect to the weak convergence topology. Moreover, it is shown by an example, that the space of probability measures on a {\sigma}-compact metric spaces not need to be {\sigma}-compact space, even though the converse statement holds true for metric spaces. The results are applied to an extended form of the distribution free newsvendor problem.