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Let G be a connected amenable group (thus, an extension of a connected normal solvable subgroup R by a connected compact group K = G R ). We show how to explicitly construct sequences { U n } of compacta in G in terms of the structural features of G which have the following property: For any “reasonable” action G × L p ( X , μ ) ↓ L p ( X , μ ) on an L p space, 1 p f ∈ L p ( X , μ ), the averages A n f= 1 |U n | ∫ U n T g −1f dg (|E|= left Haar measure in G) converge in L p norm, and pointwise μ-a.e. on X , to G -invariant functions f∗ in L p ( X , μ ). A single sequence { U n } in G works for all L p actions of G . This result applies to many nonunimodular groups, which are not handled by previous attempts to produce noncommutative generalizations of the pointwise ergodic theorem.

An individual ergodic theorem is proved for semigroups of measure-preserving transformations depending on k real parameters, which generalizes N. Wiener’s ergodic theorem.