Discontinuous translation invariant functionals

Saeki, Sadahiro

doi:10.1090/S0002-9947-1984-0728720-5

Discontinuous translation invariant functionals
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Trans. Amer. Math. Soc. 282 (1984), 403-414 Request permission

Abstract:

Let $G$ be an infinite $\sigma$-compact locally compact group. We shall study the existence of many discontinuous translation invariant linear functionals on a variety of translation invariant Fréchet spaces of Radon measures on $G$. These spaces include the convolution measure algebra $M(G)$, the Lebesgue spaces ${L^p}(G)$, where $1 \leq p \leq \infty$, and certain combinations thereof. Among other things, it will be shown that $C(G)$ has many discontinuous translation invariant functionals, provided that $G$ is amenable. This solves a problem of G. H. Meisters.

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Some problems and results on translation-invariant linear forms

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