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

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 $\mathbb{R}$.

23 pages, 2 figures