measure theory – Bochner-Minlos for moment-generating functions?

It is well-known that the Bochner-Minlos theorem characterises measures on duals of nuclear spaces by their characteristic functions. Is there a similar version for moment-generating functions?
I have a sequence of measures admitting moment-generating functions and I wish to prove something like convergence of the moment-generating functions implies convergence of the measures. But it is unclear to me, what kinds of convergences I should look at and in particular what properties I have to demand from the limiting function.