# 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

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

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

$f(t) = \int_0^\infty e^{-tx} \,dg(x),$

the integral being a Riemann–Stieltjes integral.

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

$f(t) = a + b t +\int_0^\infty (1-e^{-t x})\mu(dx)$

where $a,b \geq 0$ and $\mu$ is a measure on the positive real half-line such that

$\int_0^\infty (1\wedge x) \mu(dx) <\infty.$

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.

## 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.