LetG be a locally compact second countable abelian group, (X, μ) aσ-finite Lebesgue space, and (g, x) →gx a non-singular, properly ergodic action ofG on (X, μ). Let furthermore Γ be the character group ofG and let Sp(G, X) ⊂ Γ denote theL ∞-spectrum ofG on (X, μ). It has been shown in  that Sp(G, X) is a Borel subgroup of Γ and thatσ (Sp(G, X))<1 for every probability measureσ on Γ with lim supg→∞Re\(\hat \sigma \)(g)<1, where\(\hat \sigma \) is the Fourier transform ofσ. In this note we prove the following converse: ifσ is a probability measure on Γ with lim supg→∞Re\(\hat \sigma \)(g)<1 (g)=1 then there exists a non-singular, properly ergodic action ofG on (X, μ) withσ(Sp(G, X))=1.
KeywordsProbability Measure Abelian Group Lebesgue Space Character Group Borel Subgroup
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- 5.V. Losert and K. Schmidt,A class of probability measures arising from some problems in ergodic theory, inProbability Measures on Groups, Springer Lecture Notes in Mathematics, Vol. 706, 1979, pp. 220–238.Google Scholar
- 6.K. Schmidt,Infinite invariant measures on the circle, Symposia MathematicaXXI (1977), 37–43.Google Scholar
- 7.K. Schmidt,Mildly mixing actions of locally compact groups. II, preprint, Warwick, 1981.Google Scholar
- 8.K. Schmidt and P. Walters,Mildly mixing actions of locally compact groups. I, preprint, Warwick, 1981.Google Scholar
- 9.V. S. Varadarajan,Geometry of Quantum Theory. II, Van Nostrand, 1970.Google Scholar