Bernstein's theorem on monotone functions

In functional analysis, a branch of mathematics, Bernstein's theorem states that any real-valued function on the half-line [0, ∞) that is totally mononone is a weighted average (or expected value) of exponential functions. Total monotonicity of a function f means that

$(-1)^{n}{d^{n} \over dt^{n}}f(t)\geq 0$

for all nonnegative integers n. The "weighted average" statement can be characterized thus: there is a cumulative probability distribution function g on [0, ∞) such that

$f(t)=\int _{0}^{\infty }e^{-tx}\,dg(x).$