A Criterion for Precompactness in the Space of Hypermeasures

TL;DR

This paper introduces a new criterion for precompactness in the space of hypermeasures, extending classical measure theorems to a broader, incomplete normed space and characterizing its completion.

Contribution

It provides necessary and sufficient conditions for precompactness in the space of hypermeasures, generalizing Prokhorov's and Fernique's theorems for measures.

Findings

Criteria for precompactness similar to classical measure theorems.

Identification of the space of hypermeasures as the completion of an incomplete normed space.

Conditions depend on properties of Lipschitz functions and the underlying space.

Abstract

Let $Q$ denote the space of signed measures on the Borel $\sigma$-algebra of a separable complete space $X$. We endow $Q$ with the norm $\|q\|=\sup|\int\phi dq|$, where the supremum is taken over all Lipschitz with constant 1 functions whose module does not exceed unity. This normed space is incomplete provided $X$ is infinite and has at least one limit point. We call its completion the space of hypermeasures. Necessary and sufficient conditions for precompactness (=relative compactness) of a set of hypermeasures are found. They are similar to those of Prokhorov's and Fernique's theorems for measures.