### Abstract

This note proves a version of the pointwise ergodic theorem for functions taking values in a separable complete CAT(0)-space. The precise setting consists of an amenable locally compact group G with left Haar measure m _{G}, a jointly measurable, probability-preserving action T : G (Ω, F, P) of G on a probability space, and a separable complete CAT(0)-space (X, d) with barycentre map b. In this setting we show that if (F _{n}) _{n < 1} is a tempered Følner sequence of compact subsets of G and f : Ω → X is a measurable map such that for some (and hence any) fixed x ∈ X, we have ∫ _{Ω} d(f(ω),x) ^{2},P(dω) < ∞, then as n → ∞ the functions of empirical barycentres ω b(1/m _{G}(F _{n}) ∫ _{Fn}δ _{f}(T ^{g}ω),m _{G}(dg)) converge pointwise for almost every ω to a T-invariant function f̄: Ω → X.

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

Pages (from-to) | 145-152 |

Number of pages | 8 |

Journal | Journal of Topology and Analysis |

Volume | 3 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1 2011 |

### Fingerprint

### Keywords

- CAT(0)-space
- Pointwise ergodic theorem

### ASJC Scopus subject areas

- Analysis
- Geometry and Topology

### Cite this

*Journal of Topology and Analysis*,

*3*(2), 145-152. https://doi.org/10.1142/S1793525311000544