Invariant means on bounded vector-valued functions

Abstract

Shioji and Takahashi proved that for every bounded sequence $\{a_n\}^{\infty}_{n=0}$ of real numbers, $$\{\phi(\{a_n\}^{\infty}_{n=0}) \: | \: \phi \mathrm{\: is \: a \: Banach \: limit} \}$$ $$\: \: = \displaystyle\bigcap\limits_{j=1}^{\infty} \overline{\mathrm{co}} \{(n+1)^-1 \displaystyle\sum\limits_{k=0}^n a_{k+m} \: | \: n \geq j, m \geq 0 \}.$$ We generalize this result to bounded sequences of vectors and also apply it to bounded measurable functions.