On the Rokhlin lemma for infinite measure-preserving bijections

We study the Rokhlin lemma in the context of infinite measure-preserving bijections, and completely classify such bijections up to $\lambda$-approximate conjugacy, where $\lambda$ is the infinite measure which is preserved. This sharpens the classical version of the Rokhlin lemma, which only provides such a classification up to $\mu$-approximate conjugacy where $\mu$ is a probability measure equivalent to $\lambda$.