Some applications of set theory to measure theory and general topology

我的研究重点是一般拓扑学和测度论。有些研究带有集合论的味道，因为这两个领域偶尔都需要使用深度集合论技术。强迫方法，特别是自1963年由保罗·科恩提出证明连续统假说不能根据集合论的通常公理判定以来，已经在这些领域取得了重要进展。我的建议中涉及的一个领域是测量理论，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。我关心的问题有一种集合论的味道。过去，集合论中的技术已经成功地应用于类似的问题。我正在研究的一类问题是关于用光滑函数的图形逼近函数图形的能力。

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.