Adapted sets of measures and invariant functionals on

Author:
Rodney Nillsen

Journal:
Trans. Amer. Math. Soc. **325** (1991), 345-362

MSC:
Primary 43A15

DOI:
https://doi.org/10.1090/S0002-9947-1991-1018576-1

MathSciNet review:
1018576

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Abstract: Let be a locally compact group. If is compact, let denote the functions in having zero Haar integral. Let denote the probability measures on and let . If , let denote the subspace of generated by functions of the form , , . If is compact, . When is compact, conditions are given on which ensure that for some finite subset of , for all . The finite subset will then have the property that every -invariant linear functional on is a multiple of Haar measure. Some results of a contrary nature are presented for noncompact groups. For example, if , conditions are given upon , and upon subsets of whose elements satisfy certain growth conditions, which ensure that has discontinuous, -invariant linear functionals. The results are applied to show that for , has an infinite, independent family of discontinuous translation invariant functionals which are not -invariant.

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DOI:
https://doi.org/10.1090/S0002-9947-1991-1018576-1

Article copyright:
© Copyright 1991
American Mathematical Society