is possible to compute the measure $\sigma$ in using the analytic extension of $g$. The following result follows from [@SchillingSongVondracek12 Corollary 6.3] and the fact that $x g(x)$ is a complete Bernstein function if $g(x)$ is a Stieltjes function (see [@SchillingSongVondracek12 Theorem 6.2]).

The measure $\sigma$ in can be recovered from $g$ by the formula $$\label{eq:inversion formula}
\sigma(u,v]=-\lim_{\delta \searrow 0} \lim_{h\searrow 0} \frac{1}{\pi} \int_{u+\delta}^{v+\delta} \operatorname{Im}g(-t+ih) \, {{\mathrm d}}t, \quad 0<u<v<\infty.$$

We also record the following lemma which follows easily from Theorem \[thm:Stieltjes extensions\].

\[lemma:composition\] If $g$ is a Stieltjes function, then so is $g^\alpha$ for any $\alpha \in (0,1]$.

Next, we are interested in characterising functions with a positive Fourier transform. We refer to [@MR2284176] for the following standard results.

A function $f\colon {\mathbb R}^d \to {\mathbb C}$ is said to be *positive definite* if for every $n \in {\mathbb N}$, the $n \times n$-matrix with values $a_{ij} = f(\xi_i - \xi_j)$, $1 \leq i,j \leq n$, is positive semi-definite.

\[thm:schur\] Let $\{f_j\}_j$ be a countable family of positive definite functions. Then $\prod_{j} f_j$ is positive definite.

\[thm:bochner\] Any positive definite function continuous at zero is the Fourier transform of a finite Borel measure.

A common way of stating Bochner’s 