1. Introduction. Let (X, , μ) be a probability space, T a linear operator on ℒp
(X, , μ), for some p, 1 ≦ p ≦ ∞. Let an
be a sequence of complex numbers, n = 0, 1, …, which we shall often refer to as weights. We shall say that the weighted pointwise ergodic theorem holds for T on ℒp
, if, for every ƒ in ℒp
,
1.1
Let a denote the sequence (an). If (1.1) holds we shall say that a is Birkhoff for T on ℒp
, or, more briefly, that (a, T) is Birkhoff.
We are also interested in ergodic theorems for subsequences. Let n(k) be a subsequence. We shall say the pointwise ergodic theorem holds for the subsequence n(k) and the operator T if, for every ƒ in ℒp
,
1.2