On mixing in infinite measure spaces

Abstract

Two concepts of mixing for null-preserving transformations are introduced, both coinciding with (strong) mixing if there is a finite invariant measure. The authors believe to offer the correct answer to the old problem of defining mixing in infinite measure spaces. A sequence of sets is called semiremotely trivial if every subsequence contains a further subsequence with trivial remote σ-algebra (=tail σ-field). A transformation T is called mixing if (T−nA) is semiremotely trivial for every set A of finite measure; completely mixing if this is true for every measurable A. Thus defined mixing is exactly the condition needed to generalize certain theorems holding in finite measure case. For invertible non-singular transformations complete mixing implies the existence of a finite equivalent invariant mixing measure. If no such measure exists, complete mixing implies that for any two probability measures π1,π2, \(\pi _1 ,\pi _2 ,\pi _1 \circ T^{ - n} - \pi _2 \circ T^{ - n} \to 0\) in total variation norm.