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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Invariant means on the continuous bounded functions


Author: Joseph Rosenblatt
Journal: Trans. Amer. Math. Soc. 236 (1978), 315-324
MSC: Primary 43A07
DOI: https://doi.org/10.1090/S0002-9947-1978-0473714-8
MathSciNet review: 0473714
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Abstract: Let G be a noncompact nondiscrete $ \sigma $-compact locally compact metric group. A Baire category argument gives measurable sets $ \{ {A_\gamma }:\gamma \in \Gamma \} $ of finite measure with card $ (\Gamma ) = c$ which are independent on the open sets. One approximates $ \{ {A_\gamma }:\gamma \in \Gamma \} $ by arrays of continuous bounded functions with compact support and then scatters these arrays to construct functions $ \{ {f_\gamma }:\gamma \in \Gamma \} $ in $ {\text{CB}}(G)$ with a certain independence property. If G is also amenable as a discrete group, the existence of these independent functions shows that on $ {\text{CB}}(G)$ there are $ {2^c}$ mutually singular elements of LIM each of which is singular to TLIM.


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DOI: https://doi.org/10.1090/S0002-9947-1978-0473714-8
Article copyright: © Copyright 1978 American Mathematical Society