In real analysis, a branch of mathematics, Bernstein's theorem states that any realvalued function on the halfline [0, ∞) that is totally monotone is a mixture of exponential functions. In one important special case the mixture is a weighted average, or expected value.
Total monotonicity (sometimes also complete monotonicity) of a function f means that f is continuous on [0, ∞), infinitely differentiable on (0, ∞), and satisfies
for all nonnegative integers n and for all t > 0. Another convention puts the opposite inequality in the above definition. Completely monotone functions are also called Bernstein functions.
The "weighted average" statement can be characterized thus: there is a nonnegative finite Borel measure on [0, ∞), with cumulative distribution function g, such that
the integral being a RiemannStieltjes integral.
Further, any Bernstein function can be written in the following form:
where and μ is a measure on the positive real halfline such that
In more abstract language, the theorem characterises Laplace transforms of positive Borel measures on [0,∞). In this form it is known as the BernsteinWidder theorem, or HausdorffBernsteinWidder theorem. Felix Hausdorff had earlier characterised completely monotone sequences. These are the sequences occurring in the Hausdorff moment problem.
