Let (X, F, μ) be a finite measure space. Let T: X → X be a measure preserving transformation and let Anf denote the average of Tkf, k = 0, ..., n. Given a real positive function v on X, we prove that {Anf} converges in the a.e. sense for every f in L1(v dμ) if and only if infi ≥ 0 v(Tix) > 0 a.e., and the same condition is equivalent to the finiteness of a related ergodic power function Prf for every f in L1(v dμ). We apply this result to characterize, being T null-preserving, the finite measures ν for which the sequence {Anf} converges a.e. for every f ∈ L1(dν) and to prove that uniform boundedness of the averages in L1 is sufficient for finiteness a.e. of Pr.
@article{urn:eudml:doc:41703,
title = {Convergence of the averages and finiteness of ergodic power funtions in weighted L1 spaces.},
journal = {Publicacions Matem\`atiques},
volume = {35},
year = {1991},
pages = {465-473},
mrnumber = {MR1201568},
zbl = {0738.28012},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41703}
}
Ortega Salvador, Pedro. Convergence of the averages and finiteness of ergodic power funtions in weighted L1 spaces.. Publicacions Matemàtiques, Tome 35 (1991) pp. 465-473. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41703/