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

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 ${\operatorname {Var} _\kappa }(\mu ) = \sup \sum \nolimits _j {|\mu ({I_j}} )|/\sum \nolimits _j {\kappa (|{I_j}|)}$ (the supremum is taken over all finite partitions of ${\mathbf {T}}$ into connected subsets ${I_j}$) and ${\mu _\tau }$ denotes the $\tau$-translation of $\mu$.