In this chapter, we study the dynamics of the adjoint of a weighted composition operator, and we give necessary and sufficient conditions for this adjoint operator to be topologically transitive on the space of Radon measures on a locally compact Hausdorff space. Moreover, we provide sufficient conditions for this operator to be chaotic, and we give concrete examples. Next, we consider the real Banach space of signed Radon measures, and we give in this context the sufficient conditions for the convergence of Markov chains induced by the adjoint of an integral operator. Also, we illustrate this result by a concrete example. In addition, we obtain some structural results regarding the space of Radon measures. We characterize a class of cones whose complement is spaceable in the space of Radon measures.