Geodesic
Pointwise ergodic theorems in Lorentz spaces L(p,q) for null preserving transformations
Studia Mathematica, Tome 114 (1995) no. 3, pp. 227-236

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DOI

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 
Ryotaro Sato. Pointwise ergodic theorems in Lorentz spaces L(p,q) for null preserving transformations. Studia Mathematica, Tome 114 (1995) no. 3, pp. 227-236. doi: 10.4064/sm-114-3-227-236
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     title = {Pointwise ergodic theorems in {Lorentz} spaces {L(p,q)} for null preserving transformations},
     journal = {Studia Mathematica},
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