  The concept of a uniform set is introduced for an ergodic, measure-preserving
transformation on a non-atomic, infinite Lebesgue space. The uniform sets exist
as much as they generate the underlying $\sigma$-algebra. This leads to the
result that any ergodic, measure-preserving transformation on a non-atomic,
infinite Lebesgue space is isomorphic to a minimal homeomorphism on a locally
compact metric space which admits a unique, up to scaling, invariant Radon
measure.
