Non-trivial translation-invariant valuations on $L^\infty$

TL;DR

This paper investigates translation-invariant valuations on the space of essentially bounded functions, proving they vanish on compactly supported functions and constructing a diverse family of non-trivial valuations using ultrafilters.

Contribution

It establishes the vanishing property of translation-invariant valuations on compact support functions and introduces a novel method to construct non-trivial valuations via ultrafilters.

Findings

Valuations vanish on functions with compact support.

A rich family of non-trivial valuations is constructed.

Ultrafilters are used to generate these valuations.

Abstract

Translation-invariant valuations on the space $L^\infty(\mathbb{R}^n)$ are examined. We prove that such functionals vanish on functions with compact support. Moreover a rich family of non-trivial translation-invariant valuations on $L^\infty(\mathbb{R}^n)$ is constructed through the use of ultrafilters on $\mathbb{R}$.