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Let G be a locally compact group. If G is compact, let eo(G) denote the functions in L (G) having zero Haar integral. Let M1(G) denote the probability measures on G and let p 1(G) = M1(G) n L1(G) . If S C M1 (G), let A(Lp(G), S) denote the subspace of Lp(G) generated by functions of the form f u * f, f E Lp(G), u E 5. If G is compact, A(LP(G), S) C Lo(G). When G is compact, conditions are given on S which ensure that for some finite subset F of S, A(LP(G), F) = Lo(G) for all 1 < p < oo. The finite subset F will then have the property that every F-invariant linear functional on LP (G) is a multiple of Haar measure. Some results of a contrary nature are presented for noncompact groups. For example, if 1 < p < 00, conditions are given upon G, and upon subsets S of M1 (G) whose elements satisfy certain growth conditions, which ensure that Lp (G) has discontinuous, S-invariant linear functionals. The results are applied to show that for 1 < p < 00, LP(R) has an infinite, independent family of discontinuous translation invariant functionals which are not 3 1 (R)-invariant.

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