Integral transforms with infinitely divisible kernels

Given $r$ characteristic functions $f_1 (u), \ldots ,f_r (u)$, none of which is identically equal to one, it is shown that the integral transform
$$ \int_0^\infty \cdots \int_0^\infty {\left( {\prod\limits_{j = 1}^r {fj(u_j )^{s_j } } } \right)\,dF(s_1 , \ldots ,s_r )} $$
of the joint distribution function $F$ of $r$ non-negative random variables can be defined over a nonempty domain of natural numbers and it uniquely determines $F$. This result is used to obtain the converse of a multivariate version of a transfer theorem due to Gnedenko and Fahim, thus extending a result of Szasz and Frajeris in the univariate case. An application is also made to Lévy processes.