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

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: $$f(t) = a + bt + \int_0^\infty \left(1 - e^{-tx}\right) \mu(dx),$$ where $$a,b \geq 0$$ and $$\mu$$ is a measure on the positive real half-line such that $$\int_0^\infty \left(1\wedge x\right) \mu(dx) < \infty.$$