Some applications of set theory to analysis

我的研究重点是一般拓扑和分析，最近更具体的近似理论。有些研究带有集合论的味道，因为这两个领域偶尔需要使用深层的集合论技术。特别是自1963年以来，当它由保罗·科恩为了证明连续统假设在集合论的通常公理的基础上是不可判定的而发展出来时，强迫的方法在这些领域取得了重要的进展。Hoischen的一个定理提供了光滑函数及其衍生物的逼近的整个功能，插值封闭的离散集。这个定理有助于分析限制于真实的线的子集的序同构的整函数的存在性，这些子集在某种意义上是“粗的”，类似于巴斯-施耐德定理，它给出了限制于可数稠密集的序同构的整函数。从这项工作中获得的见解也导致了Hoischen定理 * 的变化，它包含了控制可数集上导数值的能力，同时选择 * 近似函数，以便其导数的图形是通用的，因为它们通过给定的微薄集合切割了微薄的部分。如果我们想让这些图通过一个给定的空集切割一个空 * 截面，那么在同一方向上会有更弱的结果。我们将寻求更好地理解这种设置。在进行中的工作中，我们正在研究整函数对单变量分段单调函数的共单调逼近。我的建议的另一个主题是家庭的Baire集的相互作用与措施在拓扑 * 措施空间。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一个代数同态，它从每个 * 等价类中选择一个表示，表示两个函数等价，如果它们几乎处处相等。对于完备概率空间，这些 * 首先由冯·诺依曼和马哈拉姆证明存在。它们是理解概率空间的测度代数的基础。然而，当测度不完备时，它们 * 一般不知道存在，除非在特殊的集合论 * 假设下。我建议继续以前的工作调查存在的Baire提升在各种强制扩展，也密切相关的问题 * 如何提升产品空间与提升的因素相互作用。

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.