On a pointwise ergodic theorem for multiparameter semigroups.

Sato, Ryotaro. "On a pointwise ergodic theorem for multiparameter semigroups.." Publicacions Matemàtiques 38.1 (1994): 81-87. <http://eudml.org/doc/41203>.

@article{Sato1994,
abstract = {Let Ti (i = 1, 2, ..., d) be commuting null preserving transformations on a finite measure space (X, F, μ) and let 1 ≤ p &lt; ∞. In this paper we prove that for every f ∈ Lp(μ) the averagesAnf(x) = (n + 1)-d Σ0≤ni≤n f(T1n1T2n2 ... Tdnd x)converge a.e. on X if and only if there exists a finite invariant measure ν (under the transformations Ti) absolutely continuous with respect to μ and a sequence \{XN\} of invariant sets with XN ↑ X such that νB &gt; 0 for all nonnull invariant sets B and such that the Radon-Nykodim derivative v = dν/dμ satisfies v ∈ Lq(xN,μ), 1/p + 1/q = 1, for each N ≥ 1.},
author = {Sato, Ryotaro},
journal = {Publicacions Matemàtiques},
keywords = {Teoría ergódica; Espacio de medida; Teorema de Radon-Nikodym; pointwise ergodic theorem; multiparameter semigroups; null preserving transformations; invariant measure; invariant sets; Radon-Nikodým derivative},
language = {eng},
number = {1},
pages = {81-87},
title = {On a pointwise ergodic theorem for multiparameter semigroups.},
url = {http://eudml.org/doc/41203},
volume = {38},
year = {1994},
}

TY - JOUR
AU - Sato, Ryotaro
TI - On a pointwise ergodic theorem for multiparameter semigroups.
JO - Publicacions Matemàtiques
PY - 1994
VL - 38
IS - 1
SP - 81
EP - 87
AB - Let Ti (i = 1, 2, ..., d) be commuting null preserving transformations on a finite measure space (X, F, μ) and let 1 ≤ p &lt; ∞. In this paper we prove that for every f ∈ Lp(μ) the averagesAnf(x) = (n + 1)-d Σ0≤ni≤n f(T1n1T2n2 ... Tdnd x)converge a.e. on X if and only if there exists a finite invariant measure ν (under the transformations Ti) absolutely continuous with respect to μ and a sequence {XN} of invariant sets with XN ↑ X such that νB &gt; 0 for all nonnull invariant sets B and such that the Radon-Nykodim derivative v = dν/dμ satisfies v ∈ Lq(xN,μ), 1/p + 1/q = 1, for each N ≥ 1.
LA - eng
KW - Teoría ergódica; Espacio de medida; Teorema de Radon-Nikodym; pointwise ergodic theorem; multiparameter semigroups; null preserving transformations; invariant measure; invariant sets; Radon-Nikodým derivative
UR - http://eudml.org/doc/41203
ER -