An identity for two integral transforms applied to the uniqueness of a distribution via its Laplace–Stieltjes transform

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