2016/04/03 by Dai, Xiongping
#37A05 #37A30 #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.1604.00611
Given any amenable group G (with a left Haar measure |⋅| or dg), we can select out a Følner subnet \Fθ,θ∈Θ\ from any left Følner net in G, which is L^∞-admissible, namely, for any Borel G-space (X,\mathscrX) and any φ∈ L^∞(X,\mathscrX), limθ∈Θ(1)/(|Fθ|)∫Fθφ(gx)dg=φ^*(x) ∀ x∈ X \textrmand φ^*=(gφ)^* ∀ g∈ G. Moreover, if G is σ-compact such as a locally compact second countable Hausdorff amenable group, then φ^*∈ L^∞(X,\mathscrX), φ^*(gx)=φ^*(x) a.e., and φ^* is a.e. independent of the choice of the admissible Følner net \Fθ,θ∈Θ\ in G. Consequently, we may easily obtain the ergodic disintegration of invariant probability measures for any σ-compact amenable group acting Borel on a compact metric space X by continuous transformations of X, and the existence of σ-finite invariant Radon measures for any Borel action of an amenable group on a locally compact, σ-compact, metric space X by continuous maps of X, and a L^∞-pointwise multiple ergodic theorem.