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ISSN 1088-6826(online) ISSN 0002-9939(print)



A counterexample to an F. and M. Riesz-type theorem

Author: Krzysztof Samotij
Journal: Proc. Amer. Math. Soc. 102 (1988), 337-340
MSC: Primary 30H05,; Secondary 28A12,46E99
MathSciNet review: 920996
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Abstract: A premeasure is a finitely additive complex-valued function $ \mu $ defined on the semiring of all connected subsets of $ {\mathbf{T}}$, continuous at $ \emptyset $ and with $ \mu (\emptyset ) = \mu ({\mathbf{T}}) = 0$. Let $ \kappa $ be a continuous increasing concave function on $ [0,2\pi ]$ with $ \kappa (0) = 0$. A conjecture from [3] saying that if the Poisson integral of a premeasure $ \mu $ is holomorphic in the open unit disk and $ {\operatorname{Var}_\kappa}(\mu ) < \infty $ then $ {\lim _{\tau \to 0}}{\operatorname{Var}_\kappa }({\mu _\tau } - \mu ) = 0$ is disproved, where $ {\operatorname{Var} _\kappa }(\mu ) = \sup \sum\nolimits_j {\vert\mu ({I_j}} )\vert/\sum\nolimits_j {\kappa (\vert{I_j}\vert)} $ (the supremum is taken over all finite partitions of $ {\mathbf{T}}$ into connected subsets $ {I_j}$) and $ {\mu _\tau }$ denotes the $ \tau $-translation of $ \mu $.

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Keywords: Poisson integral, analytic measures
Article copyright: © Copyright 1988 American Mathematical Society

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