Publishing house / Journals and Serials / Studia Mathematica / All issues

Abstract

Let (X,ℱ,µ) be a finite measure space and τ a null preserving transformation on (X,ℱ,µ). Functions in Lorentz spaces L(p,q) associated with the measure μ are considered for pointwise ergodic theorems. Necessary and sufficient conditions are given in order that for any f in L(p,q) the ergodic average $n^{-1} ∑^{n-1}_{i=0} f∘τ^{i}(x)$ converges almost everywhere to a function f* in $L(p_1,q_1]$, where (pq) and $(p_1,q_1]$ are assumed to be in the set ${(r,s) : r=s=1, or 1 < r < ∞ and 1 ≤ s ≤ ∞, or r = s = ∞}$. Results due to C. Ryll-Nardzewski, S. Gładysz, and I. Assani and J. Woś are generalized and unified