Suppose that $\mu_j$ are real numbers obeying the Carleman condition: $\sum_{j=0}^\infty 1/\mu_{2j}^{2j}<\infty$. Then it is well-known that in case the $\mu_j$ are positive definite, there is a unique distribution $m$ on $\mathbb{R}$ with these as the jth moment. Moreover, I can see why $m$ is a measurable function of the numbers $\mu_j$, because in fact it is even continuous from the product topology (as $j$ varies) to the vague topology, as seen via a tightness and Helly selection argument.
I am curious if any attempts have been made to make this dependence explicit. We can make it explicit if $\mu_j$ are such that the corresponding MGF exists in a neighborhood about $0$, for we can then write down the characteristic function explicitly and run the Fourier inversion formula. If integrals are "explicit" then this is good enough, and this is reasonable because the characteristic function was obtained fairly explicitly and the Fourier inversion formula is standard.
For instance, if we are very generous about what we mean by "explicit", i.e. if we allow limits, then perhaps one could approximate the sequence $\mu_j$ by a sequence for which the MGF exists, and thus get an "explicit" formula. But the problem with this is two-fold. 1. Naturally we are interested in evaluating the limit. 2. It is not obvious to me that, among sequences $\mu_j$ for which the moment problem has exactly one solution, the ones that also have finite MGF are product-topology dense.
Is this approach salvageable? Is there any other approach or pre-existing work in the literature?
