On the conservative parts of the Markov processes induced by a measurable transformation

For the two Markov processes associated with the application of a measurable transformation in a probability space in the forward and backward direction respectively, the equivalent descriptions by kernel functions and by Markov operators in L 1, L8 , and in the space of absolutely continuous finite signed measures are identified. The connections between the conservative parts of the probability space with respect to these processes and the various conservative parts associated with a measurable transformation in the literature are clarified. Finally the inclusion relations between these various conservative parts are established.