Daniel Berend

The notion of ergodicity of a measure-preserving transformation is generalized to finite sets of transformations. The main result is that if T ₁, T ₂, ..., T _s are invertible commuting measure-preserving transformations of a probability space (X, ℬ, μ) then {Mathematical expression} for any f ₁, f ₂, ..., f _s∈L ^x (X, ℬ, μ) iff T ₁×T ₂×...×T _s and all the transformations T _iT_j ¹, i≠j, are ergodic. The multiple recurrence theorem for a weakly mixing transformation follows as a special case.