On a simple proof of uniformization for continuous and discrete-state continuous-time Markov chains

Uniformization has proven to be a convenient tool for modeling and computational purposes to analyze continuous-time Markov chain applications. For the case of continuous-state space, uniformization is also intuitively obvious and most likely employed in practice. This note merely aims to show that uniformization for both the discrete- and continuous state cases is directly formalized by a simple uniqueness result from the literature