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

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 $\mathbb{R}$.

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