Some applications of set theory to analysis

点击翻译按钮获取中文摘要

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.