Some applications of set theory to analysis

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。有些研究带有集合论的味道，因为这两个领域都偶尔需要使用深度集合论技术。强迫方法，特别是自1963年由Paul Cohen提出证明连续统假设不可根据集合论的通常公理判定以来，已经在这些领域带来了重要的进展。*Hoischen的一个定理提供了整函数对光滑函数及其导数的逼近，并在闭的离散集上进行内插。与Barth-Schneider定理类似，Barth-Schneider定理给出了限制为可数稠密集的序同构的整函数，这一定理有助于分析限制为序同构的整函数的存在性。从这项工作中获得的见解也导致了Hoischen定理的变体*，该定理结合了控制可数集上的导数的值的能力，同时选择*逼近函数，使得其导数的图在某种意义上是通用的，即它们从给定的稀疏集合中切出稀疏的*部分。如果我们希望图通过给定的空集来切割空*部分，那么在相同的方向上会有较弱的结果。我们将寻求*更好地理解这种情况。在正在进行的工作中，我们正在研究一元分段单调函数与整函数的共调逼近。*我建议的另一个主题是Baire集族与拓扑*测度空间中的测度的相互作用。我正在研究概率空间上有界可测函数的代数的提升。*(提升是这个代数到自己的代数同态，它从每个*等价类中为两个函数等价的关系选择一个代表，如果它们几乎处处相等。)对于完备的概率空间，冯·诺伊曼和马哈拉姆最先证明了它们的存在。它们是理解概率空间的度量代数的基础。然而，当测量不完整时，它们*通常是未知的，除非在特殊的背景理论*假设下。我建议继续以前的工作，研究Baire提升在各种强迫扩张中的存在性，以及乘积空间的提升如何与因子的提升相互作用的密切相关问题。

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.