Mathematics

CLC number: O17

On Sufficient Conditions for the Exchangeability of Limits and Norms

极限与范数可交换性的若干充分条件研究

¹ School of Mathematics and Statistics, Liupanshui Normal University, Liupanshui 553000, Guizhou, China (六盘水师范学院　数学与统计学院，贵州　六盘水　553000)
² School of Big Data, Baoshan University, Baoshan 678000, Yunnan, China (保山学院　大数据学院，云南　保山　678000)
³ School of Mathematics and Physics, Jinggangshan University, Ji'an 343009, Jiangxi, China (井冈山大学　数理学院，江西　吉安　343009)

^† Corresponding author. E-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 11 September 2024

Abstract

In the field of mathematical analysis, the Monotone Convergence Theorem (MCT), Fatou's lemma, and the Dominated Convergence Theorem (DCT) play a crucial role in the study of the interchangeability of limits and integrals (or norms). Among them, the DCT when applied to series of numbers, namely Tannery's theorem, has been explored to a certain extent. However, the MCT and Fatou's lemma related to series of numbers still lack in-depth research in existing literature. This paper proposes and proves Fatou's lemma and the MCT in ${\mathit{\ell }}^{\mathrm{\infty }}$ space. By constructing counterexamples, it demonstrates that the DCT does not hold in ${\mathit{\ell }}^{\mathrm{\infty }}$ space. In the research of series of numbers, this paper uses the property of the infimum of real numbers to prove Fatou's lemma for series of numbers. Taking this as a theoretical foundation, it further derives the MCT and the DCT related to series of numbers. Through an in-depth analysis of the logical relationships among these three theorems, an equivalence relation among them is established, and their application values in practical problems are demonstrated through examples. In addition, based on the theory of abstract measure and integration, the proofs of the three theorems related to series of numbers are provided. It is particularly worth noting that the classical Fatou's lemma and the MCT usually require that the corresponding sequence of functions satisfies the prerequisite of being non-negative and measurable. However, Fatou's lemma and the MCT in ${\mathit{\ell }}^{\mathrm{\infty }}$ space no longer impose the non-negativity requirement on the sequence of functions. This characteristic may have potential application prospects.

摘要

在数学分析领域，单调收敛定理(MCT)、法图引理及控制收敛定理(DCT)在极限与积分(或范数)的可交换性研究中占据关键地位。其中，控制收敛定理应用于数项级数时的特定版本，即Tannery定理，已得到一定探讨；然而，与之相关的数项级数单调收敛定理及法图引理，在现有文献中缺乏深入研究。本文提出并证明了${\mathit{\ell }}^{\mathrm{\infty }}$空间下的法图引理与单调收敛定理。通过构造反例，论证了控制收敛定理在${\mathit{\ell }}^{\mathrm{\infty }}$空间中不成立。在数项级数研究方面，本文运用实数下确界的性质，证明了数项级数的法图引理，并以此为基础，进一步推导得出与数项级数相关的单调收敛定理与Tannery定理。本文分析三者之间的逻辑关联，建立起这三个定理的等价关系，并通过实例展示其在实际问题中的应用。此外，基于抽象测度与积分理论，本文还给出了基于数项级数的这三个定理的证明。特别地，经典法图引理与单调收敛定理通常要求相应函数列满足非负可测的前提条件。而${\mathit{\ell }}^{\mathrm{\infty }}$空间下的法图引理与单调收敛定理突破了这一限制，不再对函数列施加非负性要求。这一特性可能有潜在应用前景。