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An example concerning the Yosida-Hewitt decomposition of finitely additive measures

Author: Wolfgang Hensgen
Journal: Proc. Amer. Math. Soc. 121 (1994), 641-642
MSC: Primary 28A10; Secondary 28C15, 46E99
MathSciNet review: 1213861
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Abstract: Let $ \lambda $ be Lebesgue measure on the Lebesgue $ \sigma $-algebra $ \mathcal{L}$ of $ I:=]0,1[$. The author gives an example of a purely finitely additive measure $ \varphi :\mathcal{L} \to [0,1]$ vanishing on $ \lambda $-null sets such that $ \smallint f\,d\varphi = \smallint f\,d\lambda $ for every bounded continuous function f on I $ (f \in {C_b}(I))$. Consequently, $ \lambda - \varphi \in {L^\infty }(\lambda )'$ annihilates $ {C_b}(I)$ and is not purely finitely additive, contrary to an assertion of Yosida and Hewitt.

References [Enhancements On Off] (What's this?)

  • [H] W. Hensgen, Contributions to the geometry of vector-valued $ {H^\infty }$ and $ {L^1}/H_0^1$ spaces, Habilitation Thesis, Regensburg, 1992. MR 1214219 (95a:46058)
  • [HY] E. Hewitt and K. Yosida, Finitely additive measures, Trans. Amer. Math. Soc. 72 (1952), 46-66. MR 0045194 (13:543b)
  • [IT] A. and C. Ionescu-Tulcea, Topics in the theory of lifting, Ergeb. Math. Grenzgeb. (3), vol. 48, Springer, Berlin, 1969.

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Keywords: Finitely additive measures, Yosida-Hewitt decomposition
Article copyright: © Copyright 1994 American Mathematical Society

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