具有Cesàro极限的二元序列
Binary sequences with a Cesàro limit
摘要：
Cesàro极限--一个实数序列的渐近平均值--是概率论、统计学和数学分析中的一个重要算子。为了更好地理解具有Cesàro极限的序列，本文考虑了由所有具有Cesàro极限的二元序列组成的空间$\mathcal{F}$，以及相应的将每个序列映射到其Cesàro极限的泛函$ν: \mathcal{F} \rightarrow [0,1]$.列举了$\mathcal{F}$和$ν$的基本性质，并详细研究了$ν$在其上可数可加的$\mathcal{F}$中的链（全序集）.该文件的主要结果涉及的结构特性对$(\mathcal{F},ν)$，特别是$\mathcal{F}$可以分解（在一定意义上），以产生一个单调类上的$ν$是可数添加剂。在这个过程中，证明了布尔代数的单调类定理的一个轻微的推广和澄清。
英文摘要：
The Cesàro limit - the asymptotic average of a sequence of real numbers - is an operator of fundamental importance in probability, statistics and mathematical analysis. To better understand sequences with Cesàro limits, this paper considers the space $\mathcal{F}$ comprised of all binary sequences with a Cesàro limit, and the associated functional $ν: \mathcal{F} \rightarrow [0,1]$ mapping each such sequence to its Cesàro limit. The basic properties of $\mathcal{F}$ and $ν$ are enumerated, and chains (totally ordered sets) in $\mathcal{F}$ on which $ν$ is countably additive are studied in detail. The main result of the paper concerns a structural property of the pair $(\mathcal{F},ν)$, specifically that $\mathcal{F}$ can be factored (in a certain sense) to produce a monotone class on which $ν$ is countably additive. In the process, a slight generalisation and clarification of the monotone class theorem for Boolean algebras is proved.
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