# Convergence of the averages and finiteness of ergodic power funtions in weighted L1 spaces.

Publicacions Matemàtiques (1991)

- Volume: 35, Issue: 2, page 465-473
- ISSN: 0214-1493

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topOrtega Salvador, Pedro. "Convergence of the averages and finiteness of ergodic power funtions in weighted L1 spaces.." Publicacions Matemàtiques 35.2 (1991): 465-473. <http://eudml.org/doc/41703>.

@article{OrtegaSalvador1991,

abstract = {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.},

author = {Ortega Salvador, Pedro},

journal = {Publicacions Matemàtiques},

keywords = {maximal operator; convergence; weighted spaces; measure- preserving transformation; finiteness; ergodic power function; averages},

language = {eng},

number = {2},

pages = {465-473},

title = {Convergence of the averages and finiteness of ergodic power funtions in weighted L1 spaces.},

url = {http://eudml.org/doc/41703},

volume = {35},

year = {1991},

}

TY - JOUR

AU - Ortega Salvador, Pedro

TI - Convergence of the averages and finiteness of ergodic power funtions in weighted L1 spaces.

JO - Publicacions Matemàtiques

PY - 1991

VL - 35

IS - 2

SP - 465

EP - 473

AB - 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.

LA - eng

KW - maximal operator; convergence; weighted spaces; measure- preserving transformation; finiteness; ergodic power function; averages

UR - http://eudml.org/doc/41703

ER -

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