On completely additive functions related to interval-filling sequences
Let \(\Lambda\) denote the set of real sequences \(\lambda =(\lambda_n)\) with the properties: (i) \(\lambda_n>\lambda_{n+1}>0\) for any \(n\in\mathbb N\), (ii) \(L(\lambda):=\sum^{\infty}_{n=1}\lambda_n<\infty\). The sequence \(\lambda\in \Lambda\) is called interval-filling if \[ \left\{\sum^{\infty}_{n=1}\varepsilon_n\lambda_n \mid \varepsilon =(\varepsilon_1,\varepsilon_2,\ldots)\in \{0,1\}^{\mathbb N}\right\}=[0,L(\lambda)]. \] Let \(\text{IF}\) denote the set of interval-filling sequences. For \(\lambda\in \text{IF}\) the function \(F: [0,L(\lambda)]\to\mathbb R\) is called completely additive with respect to \(\lambda\) if \(F(\sum^{\infty}_{n=1}\varepsilon_n\lambda_n)=\sum^{\infty}_{n=1}\varepsilon_n F(\lambda_n)\) holds for any sequence \(\varepsilon =(\varepsilon_ 1,\varepsilon_ 2,\ldots)\in \{0,1\}^{\mathbb N}\). In the present paper the authors prove the following main result: If \(\lambda\in \text{IF}\) and \(F: [0,L(\lambda)]\to\mathbb R\) is a completely additive function with respect to \(\lambda\), then there exists \(c\in\mathbb R\), such that \(F(x)=cx\) holds for any \(x\in [0,L(\lambda)]\).
scientific article; zbMATH DE number 4156512