The Bijection Property in the Law of Total Probability and Its Application in Communication Theory

The bijection property in the law of total probability has potential applications. For instance, it relates to the uniqueness of a capacity-achieving distribution, resulting in maximum information transmission from a transmitter. This letter focuses on this property and demonstrates, through a counterexample, that it does not always exist. Additionally, the letter establishes a link between the law and the moment problem context by applying Taylor expansion to the integrand. As a main result, this research provides a sufficient condition for a unique measure to be satisfied in the law of total probability, named generalized Carleman’s condition. Lastly, it applies this condition when the integrand is the generalized Poisson distribution, which is a standard model typically used for counting events in science.