A characterization of probability measures by f-moments

Given a real-valued continuous function ƒ on the half-line [0,∞) we denote by P*(ƒ) the set of all probability measures μ on [0,∞) with finite ƒ-moments ${\displaystyle {ʃ}_0^{\infty }ƒ\left(x\right){\mu }^{\cdot n}\left(dx\right)}$ (n = 1,2...). A function ƒ is said to have the identification property} if probability measures from P*(ƒ) are uniquely determined by their ƒ-moments. A function ƒ is said to be a Bernstein function} if it is infinitely differentiable on the open half-line (0,∞) and ${\displaystyle {(-1)}^n{ƒ}^{(n+1)}\left(x\right)}$ is completely monotone for some nonnegative integer n. The purpose of this paper is to give a necessary and sufficient condition in terms of the representing measures for Bernstein functions to have the identification property.