Mathematical Sciences: Applications of Nonstandard Analysis to Measure Theory, Potential Theory & Related Topics

勒布将继续他的工作，将非标准模型应用于测量理论、位势理论和随机过程。该方法使用在非标准点集上定义的测量空间。这些空间使有限组合方法在无限概率空间的研究中成为可能。勒布将把这种测量理论与非标准船体理论结合起来。他还将继续他在势能理论中表示度量和理想边界的工作。本研究项目属于非标准分析领域。这门课是在一个实数的非标准模型的基础上，试图发展标准的分析主题，包括积分和微分。在非标准模型中，人们有“无限小”数字的概念，它形式化了极限概念背后的直观概念。这种形式构造通常会导致更高的组合效率。

Loeb will continue his work in applying nonstandard models to measure theory, potential theory, and stochastic processes. This approach uses measure spaces defined on nonstandard point sets. These spaces make finite combinatorial methods available in research on infinite probability spaces. Loeb will combine this measure theory with the theory of nonstandard hulls. He also will continue his work on representing measures and ideal boundaries in potential theory. This research project is in the area of nonstandard analysis. This subject arises from the attempt to develop the standard topics of analysis, including integration and differentiation, based on a nonstandard model for the real numbers. In the nonstandard model, one has the notion of "infinitely small" numbers which formalize the intuitive notions behind the concept of limit. This formal construct often leads to greater combinatorial efficiency.