  It is well known that the Laplace-Stieltjes transform of a nonnegative random
variable (or random vector) uniquely determines its distribution function. We
extend this uniqueness theorem by using the Muntz-Szasz Theorem and the
identity for the Laplace-Stieltjes and Laplace-Carson transforms of a
distribution function. The latter appears for the first time to the best of our
knowledge. In particular, if X and Y are two nonnegative random variables with
joint distribution H, then H can be characterized by a suitable set of
countably many values of its bivariate Laplace-Stieltjes transform. The general
high-dimensional case is also investigated. Besides, Lerch's uniqueness theorem
for conventional Laplace transforms is extended as well. The identity can be
used to simplify the calculation of Laplace-Stieltjes transforms when the
underlying distributions have singular parts. Finally, some examples are given
to illustrate the characterization results via the uniqueness theorem.
