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.
Citation
Yuan-Chuan LI. "Invariant means on bounded vector-valued functions." J. Math. Soc. Japan 63 (3) 819 - 836, July, 2011. https://doi.org/10.2969/jmsj/06330819
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