Invariant means on bounded vector-valued functions

Abstract

Shioji and Takahashi proved that for every bounded sequence {a_n}_n=0^∞ of real numbers,
{ φ({a_n}_n=0^∞) | φ is a Banach limit}
= ∩_j=1^∞$¥overline{co}$ {(n+1)^-1 Σ_k=0ⁿ a_k+m | n≥j, m≥0 }.
We generalize this result to bounded sequences of vectors and also apply it to bounded measurable functions.