A cat(0)-valued pointwise ergodic theorem

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) ∫ _Fnδ _f(T ^gω),m _G(dg)) converge pointwise for almost every ω to a T-invariant function f̄: Ω → X.