Some applications of set theory to measure theory and general topology

我的研究主要集中在一般拓扑学和测度理论。一些研究具有集合论的味道，因为这两个领域偶尔需要使用深层的集合论技术。特别是强迫的方法，自1963年以来，它是由保罗·科恩（Paul Cohen）开发的，用于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，导致了这些领域的重要进展。在我的建议中涉及的一个领域是测度论，它致力于研究计算长度、面积和体积的方法以及类似类型的更一般的概念。我所关心的问题有一种集合论的味道。集合论的技术在过去已经成功地应用于类似的问题。我正在研究的问题之一是关于用光滑函数图近似函数图的能力。

My research focuses on general topology and measure 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. One area dealt with in my proposal is measure theory, which is devoted to the study of methods for calculating lengths, areas and volumes and more general notions of a similar type. The problems that concern me have a set-theoretic flavor. Techniques from set theory have been successfully applied to similar problems in the past. One of the families of problems I am studying concerns the ability to approximate graphs of functions by graphs of smooth functions.