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

Abstract

Let G G be a locally compact group. If G G is compact, let L 0 p ( G ) L_0^p(G) denote the functions in L p ( G ) {L^p}(G) having zero Haar integral. Let M 1 ( G ) {M^1}(G) denote the probability measures on G G and let P 1 ( G ) = M 1 ( G ) ∩ L 1 ( G ) {\mathcal {P}^1}(G) = {M^1}(G) \cap {L^1}(G) . If S ⊆ M 1 ( G ) S \subseteq {M^1}(G) , let Δ ( L p ( G ) , S ) \Delta ({L^p}(G),S) denote the subspace of L p ( G ) {L^p}(G) generated by functions of the form f − μ ∗ f f - \mu \ast f , f ∈ L p ( G ) f \in {L^p}(G) , μ ∈ S \mu \in S . If G G is compact, Δ ( L p ( G ) , S ) ⊆ L 0 p ( G ) \Delta ({L^p}(G),S) \subseteq L_0^p(G) . When G G is compact, conditions are given on S S which ensure that for some finite subset F F of S S , Δ ( L p ( G ) , F ) = L 0 p ( G ) \Delta ({L^p}(G),F) = L_0^p(G) for all 1 > p > ∞ 1 > p > \infty . The finite subset F F will then have the property that every F F -invariant linear functional on L p ( G ) {L^p}(G) is a multiple of Haar measure. Some results of a contrary nature are presented for noncompact groups. For example, if 1 ≤ p ≤ ∞ 1 \leq p \leq \infty , conditions are given upon G G , and upon subsets S S of M 1 ( G ) {M^1}(G) whose elements satisfy certain growth conditions, which ensure that L p ( G ) {L^p}(G) has discontinuous, S S -invariant linear functionals. The results are applied to show that for 1 ≤ p ≤ ∞ 1 \leq p \leq \infty , L p ( R ) {L^p}(\mathbb {R}) has an infinite, independent family of discontinuous translation invariant functionals which are not P 1 ( R ) {\mathcal {P}^1}(\mathbb {R}) -invariant.