Note on Completely Monotone Densities

In 2 it is proved that mixtures of exponential distributions are infinitely divisible (id). In 3 it is proved that the same holds for the discrete analogue, i. e. for mixtures of geometric distributions. In this note we show that these results imply that a density function f (x) (or distribution \pₙ\ on the integers) is id if the function f (x) (or the sequence \pₙ\ is completely monotone (cm). For the definition and properties of cm functions and sequences we refer to 1.