# Bernstein's theorem on monotone functions

In real analysis, a branch of mathematics, **Bernstein's theorem** states that every real-valued function on the half-line [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.

The "weighted average" statement can be characterized thus: there is a non-negative finite Borel measure on [0, ∞) with cumulative distribution function *g* such that

the integral being a Riemann–Stieltjes integral.

In more abstract language, the theorem characterises Laplace transforms of positive Borel measures on [0, ∞). In this form it is known as the **Bernstein–Widder theorem**, or **Hausdorff–Bernstein–Widder theorem**. Felix Hausdorff had earlier characterised completely monotone sequences. These are the sequences occurring in the Hausdorff moment problem.

## Bernstein functions

Nonnegative functions whose derivative is completely monotone are called *Bernstein functions*. Every Bernstein function has the Lévy–Khintchine representation:

where and is a measure on the positive real half-line such that

## References

- S. N. Bernstein (1928). "Sur les fonctions absolument monotones".
*Acta Mathematica*.**52**: 1–66. doi:10.1007/BF02592679. - D. Widder (1941).
*The Laplace Transform*. Princeton University Press. - Rene Schilling, Renming Song and Zoran Vondracek (2010).
*Bernstein functions*. De Gruyter.