Abstract

Let \(\mathbf{E}=\mathbf{E}(\Omega,\mathcal{F},\mu)\) be a symmetric Banach space of measurable functions on a measure space \((\Omega,\mathcal{F},\mu)\). We prove a version of Mean (Statistical) Ergodic Theorem for Cesáro averages \(A_{n,T}f=1/n\sum_{k=1}^{n}T^{k-1}f\), \(f\in\mathbf{E}\), while operators on \(\mathbf{E}\) are induced by positive absolute contraction in \(\mathbf{L}_{1}+\mathbf{L}_{\infty}=(\mathbf{L}_{1}+\mathbf{L}_{\infty})(\Omega,\mathcal{F},\mu)\).

中文翻译：

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可测函数对称空间中的平均遍历定理

摘要

令\(\mathbf{E}=\mathbf{E}(\Omega,\mathcal{F},\mu)\)是一个测量空间\((\Omega,\mathcal{ F},\mu)\)。我们证明了一个版本的切萨罗平均值的平均（统计）遍历定理\(A_{n,T}f=1/n\sum_{k=1}^{n}T^{k-1}f\) , \ (f\in\mathbf{E}\)，而\(\mathbf{E}\)上的运算符是由\(\mathbf{L}_{1}+\mathbf{L}_{ 中的正绝对收缩引起的\infty}=(\mathbf{L}_{1}+\mathbf{L}_{\infty})(\Omega,\mathcal{F},\mu)\)。