The graph-related symmetries of a reaction network give rise to certain special equilibria (such as complex balanced equilibria) in deterministic models. Correspondingly, in stochastic models these symmetries give rise to certain special stationary measures. Previous work by Anderson, Craciun and Kurtz identified stationary distributions of a complex balanced network; meanwhile Cappelletti and Wiuf developed the notion of complex balancing for stochastic systems. We define and establish the relations between reaction balanced measure, complex balanced measure and reaction vector balanced measure and prove that with mild additional hypotheses, the former two are stationary distributions. We develop the idea of decomposing both deterministic and stochastic systems into so-called ``factor systems'' and we establish the correspondence between factors of a complex balanced deterministic system and those of the corresponding stochastic system. Furthermore, in spirit of earlier work by Joshi, we give additional conditions for when detailed balance of Markov chain theory implies detailed balance of reaction network theory.