Measure-valued continuous curves and processes in total variation norm

Several equivalent characterizations of continuous curves in the total variation norm are given, which enable us to provide a sufficient condition for a purely atomic finite measure-valued stochastic process to possess a version with continuous sample paths in the total variation norm. Our criterion is in the form of Kolmogorov’s continuity theorem. As an application, we apply this criterion to study the sample path property of finite measure-valued diffusions with immigrations constructed by Shiga (1990).