### Abstract

Let (Ω , μ) be a σ-finite measure space, and let X⊂ L
^{1}
(Ω) + L
^{∞}
(Ω) be a fully symmetric space of measurable functions on (Ω , μ). If μ(Ω) = ∞, necessary and sufficient conditions are given for almost uniform convergence in X (in Egorov’s sense) of Cesàro averages Mn(T)(f)=1n∑k=0n-1Tk(f) for all Dunford–Schwartz operators T in L
^{1}
(Ω) + L
^{∞}
(Ω) and any f∈ X. If (Ω , μ) is quasi-non-atomic, it is proved that the averages M
_{n}
(T) converge strongly in X for each Dunford–Schwartz operator T in L
^{1}
(Ω) + L
^{∞}
(Ω) if and only if X has order continuous norm and L
^{1}
(Ω) is not contained in X.

Original language | English (US) |
---|---|

Pages (from-to) | 229-253 |

Number of pages | 25 |

Journal | Acta Mathematica Hungarica |

Volume | 157 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1 2019 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

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*Acta Mathematica Hungarica*, vol. 157, no. 1, pp. 229-253. https://doi.org/10.1007/s10474-018-0872-1

**Almost uniform and strong convergences in ergodic theorems for symmetric spaces.** / Chilin, V.; Litvinov, Semyon.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Almost uniform and strong convergences in ergodic theorems for symmetric spaces

AU - Chilin, V.

AU - Litvinov, Semyon

PY - 2019/2/1

Y1 - 2019/2/1

N2 - Let (Ω , μ) be a σ-finite measure space, and let X⊂ L 1 (Ω) + L ∞ (Ω) be a fully symmetric space of measurable functions on (Ω , μ). If μ(Ω) = ∞, necessary and sufficient conditions are given for almost uniform convergence in X (in Egorov’s sense) of Cesàro averages Mn(T)(f)=1n∑k=0n-1Tk(f) for all Dunford–Schwartz operators T in L 1 (Ω) + L ∞ (Ω) and any f∈ X. If (Ω , μ) is quasi-non-atomic, it is proved that the averages M n (T) converge strongly in X for each Dunford–Schwartz operator T in L 1 (Ω) + L ∞ (Ω) if and only if X has order continuous norm and L 1 (Ω) is not contained in X.

AB - Let (Ω , μ) be a σ-finite measure space, and let X⊂ L 1 (Ω) + L ∞ (Ω) be a fully symmetric space of measurable functions on (Ω , μ). If μ(Ω) = ∞, necessary and sufficient conditions are given for almost uniform convergence in X (in Egorov’s sense) of Cesàro averages Mn(T)(f)=1n∑k=0n-1Tk(f) for all Dunford–Schwartz operators T in L 1 (Ω) + L ∞ (Ω) and any f∈ X. If (Ω , μ) is quasi-non-atomic, it is proved that the averages M n (T) converge strongly in X for each Dunford–Schwartz operator T in L 1 (Ω) + L ∞ (Ω) if and only if X has order continuous norm and L 1 (Ω) is not contained in X.

UR - http://www.scopus.com/inward/record.url?scp=85053791001&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85053791001&partnerID=8YFLogxK

U2 - 10.1007/s10474-018-0872-1

DO - 10.1007/s10474-018-0872-1

M3 - Article

AN - SCOPUS:85053791001

VL - 157

SP - 229

EP - 253

JO - Acta Mathematica Hungarica

JF - Acta Mathematica Hungarica

SN - 0236-5294

IS - 1

ER -