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 Müntz–Szász 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.

众所周知，非负随机变量（或随机向量）的拉普拉斯-斯蒂尔捷斯变换唯一确定其分布函数。我们通过使用 Müntz-Szász 定理和分布函数的 Laplace-Stieltjes 和 Laplace-Carson 变换的恒等式来扩展这个唯一性定理。据我们所知，后者是第一次出现。特别地，如果X和Y是两个具有联合分布H 的非负随机变量，则H可以用一组合适的可数多个值来表征其二元 Laplace-Stieltjes 变换。还研究了一般的高维情况。此外，还扩展了传统拉普拉斯变换的 Lerch 唯一性定理。当基础分布具有奇异部分时，恒等式可用于简化 Laplace-Stieltjes 变换的计算。最后，给出了一些例子来说明通过唯一性定理的表征结果。