Invariant measures with values in normed A-modules. III. (Spanish: Medidas Invariantes con valores en A-módulos normados III)

Abstract

The set of all continuous functions with compact supports from a locally compact topological group G to a normed A-module X (A being a normed ring) is denoted by K(G,X). In this work the author characterizes all A-linear maps μ:K(G,X)→X satisfying the two conditions stated below: (1) for compact K⊂G, there exists a positive constant MK such that ∥μ(f)∥≤MKsups∈G∥f(s)∥ for all f∈K(G,K) with Supp f⊂K; (2) μ(sf)=μ(f), s∈G, for all f∈K(G,X), where sf is the function defined by sf(t)=f(s−1t). Theorems of the following type that generalize the uniqueness theorem for Haar measure are also obtained: There exists a μ:K(G,X)→X satisfying conditions (1) and (2) such that every ν of the same kind has the form T∘μ, where T is a bounded A-linear map from X to X. These results are easily generalized to the case in which X is a locally convex Hausdorff topological vector space over R or C.