向量格中的巴拿赫原理
Banach's principle in vector lattices
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中文总结 AI 辅助
该研究在向量格中发展抽象巴拿赫原理，推导遍历定理的格论版本，将框架应用于$L^0(\Omega)$空间，重新考察经典相关定理。
中文摘要 AI 辅助
我们在向量格中发展了抽象巴拿赫原理，并将其应用于获得个体遍历定理和极大遍历定理的格论版本，无需借助任何测度表示。考虑取值于戴德金σ-完全向量格的有界算子序列，该向量格赋予满足σ-勒贝格性质的局部实心拓扑，我们证明该序列序收敛的点集是闭子空间，且只要在稠密子集上收敛，该点集就与整个空间重合。研究序连续巴拿赫格上的正的幂有界平均遍历算子时，我们由严格正的序连续泛函在泛完备化上构造一个拓扑，证明切萨罗平均在泛完备化中序收敛，尤其在原格中uo-收敛。此外，我们引入超不变对的概念，通过带投影推导出格论的霍普夫不等式及弱型估计，由此得到抽象极大遍历定理。该定理的谱论版本可由经典佩龙-弗罗贝尼乌斯理论得到。最后，我们将抽象框架专门化到模型空间$L^0(\Omega)$，重新考察经典巴拿赫原理、霍普夫-邓福德-施瓦茨定理及杜布鞅收敛定理。
英文摘要
We develop an abstract Banach principle in vector lattices and apply it to obtain lattice-theoretic versions of the individual and maximal ergodic theorems, without recourse to any measure representation. Considering a sequence of bounded operators with values in a Dedekind $σ$-complete vector lattice endowed with a locally solid topology satisfying the $σ$-Lebesgue property, we prove that the set of points at which the sequence is order convergent is a closed subspace and coincides with the whole space whenever convergence holds on a dense subset. Investigating positive, power-bounded, mean ergodic operators on order continuous Banach lattices, we construct a topology on the universal completion induced by a strictly positive order continuous functional and prove that the Cesàro means converge in order in the universal completion and, in particular, uo-converge in the original lattice. Moreover, we introduce the notion of a superinvariant pair and derive a lattice-theoretic Hopf inequality together with a weak type estimate via band projections, which yields an abstract maximal ergodic theorem. A spectral-theoretic version of the theorem follows from classical Perron--Frobenius theory. Finally, we specialise the abstract framework to the model space $L^0(Ω)$ and revisit the classical Banach principle, the Hopf--Dunford--Schwartz theorem and Doob's martingale convergence theorem.
