Some applications of set theory to analysis

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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.