and positive*.

\[thm:HBW\] Let $g$ be completely monotone. Then it is the Laplace transform of a unique Borel measure $\mu$ on $[0,\infty)$, i.e. $$\label{eq:representation of completely monotone functions}
g(\lambda)={\mathcal{L}}(\mu; \lambda):=\int_{[0,\infty)} \exp(-\lambda s) \, {{\mathrm d}}\mu(s).$$ Conversely, if $\mu$ is a Borel measure on $[0,\infty)$ with ${\mathcal{L}}(\mu;\lambda)<\infty$ for every $\lambda>0$, then $\lambda \mapsto 
{\mathcal{L}}(\mu;\lambda)$ is a completely monotone function.

For a proof of this result, see [@SchillingSongVondracek12 Theorem 1.4]. A consequence of Bernstein’s theorem is that if $g$ is completely monotone, then holds with strict inequality for every $\lambda$ and every $n$, unless $g$ is identically constant. Note also that the measure $\mu$ in is finite if and only if $\lim_{\lambda\searrow 0} g(\lambda)<\infty$.

For later use we introduce the following subclass of the completely monotone functions.

\[def:Stieltjes function\] A function $g\colon (0, \infty)\to [0,\infty)$ is called a (nonnegative) [*Stieltjes function*]{} if it can be written in the form $$\label{eq:Stieltjes}
g(\lambda)=\frac{a}{\lambda}+b+\int_{(0,\infty)} \frac{1}{\lambda+t}\, {{\mathrm d}}\sigma(t),$$ where $a, b\ge 0$ are constants and $\sigma$ is a Borel measure on $(0, \infty)$ such that $
\int_{(0,\infty)} \frac{{{\mathrm d}}\sigma(t)}{1+t}<\infty$.

Note that if $g$ has a finite limit at the origin, then $a=0$ and $\int_{(0,\infty)} \frac{{{\mathrm d}}\sigma(t)}{t}<\infty$ by Fatou’s lemma. Moreover, $b=\lim_{\lambda \to \infty} g(\lambda)$. The fact 