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Transactions of the American Mathematical Society

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Adapted sets of measures and invariant functionals on $ L\sp p(G)$


Author: Rodney Nillsen
Journal: Trans. Amer. Math. Soc. 325 (1991), 345-362
MSC: Primary 43A15
DOI: https://doi.org/10.1090/S0002-9947-1991-1018576-1
MathSciNet review: 1018576
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Abstract: Let $ G$ be a locally compact group. If $ G$ is compact, let $ L_0^p(G)$ denote the functions in $ {L^p}(G)$ having zero Haar integral. Let $ {M^1}(G)$ denote the probability measures on $ G$ and let $ {\mathcal{P}^1}(G) = {M^1}(G) \cap {L^1}(G)$. If $ S \subseteq {M^1}(G)$, let $ \Delta ({L^p}(G),S)$ denote the subspace of $ {L^p}(G)$ generated by functions of the form $ f - \mu\ast f$, $ f \in {L^p}(G)$, $ \mu \in S$. If $ G$ is compact, $ \Delta ({L^p}(G),S) \subseteq L_0^p(G)$ . When $ G$ is compact, conditions are given on $ S$ which ensure that for some finite subset $ F$ of $ S$, $ \Delta ({L^p}(G),F) = L_0^p(G)$ for all $ 1 < p < \infty $. The finite subset $ F$ will then have the property that every $ F$-invariant linear functional on $ {L^p}(G)$ is a multiple of Haar measure. Some results of a contrary nature are presented for noncompact groups. For example, if $ 1 \leq p \leq \infty $, conditions are given upon $ G$, and upon subsets $ S$ of $ {M^1}(G)$ whose elements satisfy certain growth conditions, which ensure that $ {L^p}(G)$ has discontinuous, $ S$-invariant linear functionals. The results are applied to show that for $ 1 \leq p \leq \infty$, $ {L^p}(\mathbb{R})$ has an infinite, independent family of discontinuous translation invariant functionals which are not $ {\mathcal{P}^1}(\mathbb{R})$-invariant.


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DOI: https://doi.org/10.1090/S0002-9947-1991-1018576-1
Article copyright: © Copyright 1991 American Mathematical Society

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