Moments of discrete measures with dense jumps induced by β-expansions

Let β>1. Through an appeal to β-expansions we define a strictly increasing and left-continuous function μβ on [0,1]. Then μβ turns out to be a pure jump distribution. In other words, its associated Lebesgue-Stieltjes measure is discrete, i.e., a summation of point masses. The present note studies the moment of this discrete measure and its asymptotics.