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

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.