New uniqueness conditions for the classical moment problem (Q1946447)

According to the author, in a symmetric space \(E\) on (0,1] (Marcinkiewicz and Orlicz spaces are symmetric) there is \textit{uniqueness in the Hamburger moment problem} if the conditions \(x\in E\) and \(\int_0^1x(t)^ndt=\int_0^1y(t)^ndt\) for all integers \(n\geq1\) imply \(\mu\{t:x(t)>\tau\}=\mu\{t:y(t)>\tau\}\) for all real numbers \(\tau\), where \(\mu\) is the Lebesgue measure. The author characterizes some Marcinkiewicz and Orlicz spaces having this property. Similar results are given, related to the \textit{uniqueness in the Stieltjes moment problem}. Some ideas from the theory of extrapolation, due to \textit{B. Jawerth} and \textit{M. Milman} [Mem. Am. Math. Soc. 440 (1991; Zbl 0733.46040)], are used.