Some applications of set theory to analysis

集合论在分析中的一些应用

• 批准号: RGPIN-2018-05498
• 负责人: Burke, Maxim
• 金额: $ 1.17万
• 依托单位: University of Prince Edward Island
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=713710
• 关键词: Some; applications; set; theory; analysis

My research focuses on general topology and analysis, recently more specifically approximation theory. Some of the research has a set-theoretic flavor, as both of these areas occasionally require the use of deep set-theoretic techniques. The method of forcing especially, since 1963, when it was developed by Paul Cohen for proving that the Continuum Hypothesis is not decidable on the basis of the usual axioms for set theory, has led to important advances in these areas. A theorem of Hoischen provides approximation of smooth functions and their derivatives by entire functions, with interpolation on a closed discrete set. This theorem has been useful in helping to analyze the existence of entire functions restricting to order-isomorphisms of subsets of the real line which are 'thick' in a sense, analogous to the Barth-Schneider theorem, which gives entire functions restricting to order-isomorphisms of countable dense sets. The insights gained from this work have also led to variations on the Hoischen theorem that incorporate the ability to control the values of the derivatives on a countable set while choosing the approximating function so that the graphs of its derivatives are generic in the sense that they cut a meager section through a given meager set. If we want the graphs to instead cut a null section through a given null set, then there are weaker results in the same direction. We will seek to better understand this setting. In work in progress, we are examining the comonotone approximation of piecewise monotone functions of one variable by entire functions. Another theme of my proposal is the interaction of the family of Baire sets with the measure in a topological measure space. I am studying liftings for the algebra of bounded measurable functions on a probability space. (A lifting is an algebra homomorphism of this algebra into itself which selects a representative from each equivalence class for the relation that two functions are equivalent if they are equal almost everywhere.) These were first shown to exist, for complete probability spaces, by von Neumann and Maharam. They are fundamental to understanding the measure algebra of a probability space. When the measure is not complete however, they are in general not known to exist, except under special settheoretic assumptions. I propose to continue previous work investigating the existence of Baire liftings in various forcing extensions, and also closely related question of how liftings for product spaces interact with liftings for the factors.

我的研究重点是一般拓扑学和分析，最近更具体地说是逼近理论。其中一些 这项研究有一种集合论的味道，因为这两个领域偶尔都需要使用深度集合论 技巧。强迫的方法，尤其是自1963年以来，当保罗·科恩为了证明 在集合论的通常公理的基础上，连续统假设是不可判定的，导致了重要的 这些领域的进展。 Hoischen的一个定理提供了光滑函数及其导数的整函数逼近，并在闭离散集上进行内插。这个定理在帮助分析 整函数的存在性限制为实直线的子集的序同构，在某种意义上是‘粗’的， 类似于Barth-Schneider定理，它给出了整函数限制为 可数稠密集。从这项工作中获得的洞察力也导致了对Hoischen定理的变化 包含控制可数集上的导数的值的能力，同时选择 逼近函数，使得它的导数的图形在某种意义上是通用的，即它们切了一个很少的 部分通过一个给定的稀疏集合。如果我们想让图表改为截取一个空 部分通过给定的空集，则在同一方向上存在较弱的结果。我们将寻求 以更好地理解这一设置。在进行中的工作中，我们 研究了一元分片单调函数与整函数的同调逼近。 我的提议的另一个主题是Baire集族与拓扑图中的度量的相互作用 测量空间。我正在研究概率空间上有界可测函数代数的提升。 (提升是这个代数到自身的代数同态，它从每个代数中选择一个表示 如果两个函数几乎处处相等，则两个函数等价的关系的等价类。这些 对于完备的概率空间，冯·诺伊曼和马哈拉姆首先证明了它们的存在。它们是基本的 来理解概率空间的度量代数。然而，当措施尚未完成时，他们 通常是未知的，除非在特殊的集合理论下 假设。我建议继续以前的 研究Baire提升在各种强迫扩张中的存在性的工作，以及与之密切相关的问题 产品空间的提升如何与因素的提升相互作用。