Concrete examples of transformations which preserve a finite measure and ones which preserve an infinite measure exist in the literature. It's known that a basic difference exists between transformations which preserve a finite measure and those which preserve an infinite measure (see, e.g., [A. B. Hajian and S. Kakutani, Weakly wandering sets and invariant measures,
Trans. Amer. Math. Soc. 110 (1964), 136-151]). In the context of this question, see also [ https://www.researchgate.net/publication/243077292_Weakly_Wandering_Vectors_and_Weakly_Independent_Partitions] and [https://www.researchgate.net/publication/268890175_Characterizations_of_measurability-preserving_ergodic_transformations].
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