Invariant measures with values in locally convex spaces. (Spanish: Medidas invariantes con valores en espacios localmente convexos)

Abstract

Let E be a locally compact space, and X a locally convex (real or complex) Hausdorff quasicomplete vector space. Let μ0 be a positive Radon measure on E; corresponding to this measure the author defines a certain measure μ on E with values on X. In the case in which E is a locally compact topological group, and μ0 a left [right] Haar measure, μ is also a left [right] Haar measure. Let T:X !X be a continuous linear mapping, and μ a left [right] Haar measure on E with values on X; then T ·μ is also a left [right] Haar measure. Conversely, let μ be a left [right] Haar measure on E with values on X, let be any left [right] Haar measure on E with values on X; the author proves that = T · μ, where T:X ! X is a continuous linear mapping. This generalizes the known theorem of H. Weyl on positive Haar measures.