Application of the Krein-Milman theorem to completely monotonic functions

Abstract

A real valued function f on (0, ∞) is said to be completely monotonic if f has derivatives f ⁽⁰⁾ = f, f ⁽¹⁾, f ⁽²⁾,... of all orders and if (-1)ⁿ f ⁽ⁿ⁾ ≧ 0 for n = 0, 1, 2,... Thus, f is nonnegative and non-increasing, as is each of the functions (-1)ⁿ f ⁿ. [[Some examples: x ^-α and e^αx (α ≧ 0).] S. Bernstein proved a fundamental representation theorem for such functions. (See [82] for several proofs and much related material). We will prove the theorem only for bounded functions; the extension to unbounded functions (with infinite repre-senting measures) follows from this by classical arguments [82]. We denote the one-point compactification of [0, ∞) by [0, ∞].