Journal of the Mathematical Society of Japan

Invariant means on bounded vector-valued functions

Yuan-Chuan LI

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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.

Article information

J. Math. Soc. Japan, Volume 63, Number 3 (2011), 819-836.

First available in Project Euclid: 1 August 2011

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Zentralblatt MATH identifier

Primary: 40G05: Cesàro, Euler, Nörlund and Hausdorff methods 47A35: Ergodic theory [See also 28Dxx, 37Axx]
Secondary: 40E05: Tauberian theorems, general

Cesáro limit Abel limit mean Banach limit σ-limit weakly almost convergent strongly almost convergent ergodic net semi-ergodic net


LI, Yuan-Chuan. Invariant means on bounded vector-valued functions. J. Math. Soc. Japan 63 (2011), no. 3, 819--836. doi:10.2969/jmsj/06330819.

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