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

勒布教授的项目涉及在实际分析和相关领域使用非标准模型。他的方法的基本原理是使用定义在非标准点集上的标准度量空间。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。他工作的另一个方面涉及向量格的非标准壳(由非标准向量构成的标准无限维向量空间)及其在测度论和积分中的作用。他还将继续研究他为一般势能理论开发的类似于马丁边界的理想边界。所有的数学分析都建立在实数系统的结构上，实数系统被标识为双向无限直线上的点。在这个系统中，每个非零的实数都离零有一个有限的正距离，这是常识和基本原理，但有时会有一种强大的智力诱惑，去考虑数字无限--无限接近于零，或者无限远。屈从于这种诱惑，人们可以构建一个连贯的系统，称为非标准实数，它包括无穷小和无穷大，并由此进行非标准分析。勒布教授的研究涉及从标准分析中的问题开始，例如在概率论和数学物理中出现的问题，使用非标准方法来获得一些空间，然后回到标准项下的原始问题的解决方案。

Professor Loeb's project involves the use of nonstandard models in real analysis and related areas. Fundamental in his approach is the use of standard measure spaces defined on nonstandard point sets. These spaces make finite combinatorial methods available in research on infinite probability spaces. Another facet of his work deals with nonstandard hulls (standard, infinite-dimensional vector spaces formed from nonstandard ones) of vector lattices and their role in measure theory and integration. He will also continue research on an ideal boundary, analogous to the Martin boundary, that he has developed for general potential theories. All of mathematical analysis is built upon the structure of the system of real numbers, which are identified as points on a two-way infinite line. It is commonsensical and fundamental in this system that every non-zero real number lies at a finite positive distance away from zero, but sometimes there is a powerful intellectual temptation to consider numbers infinites- imally close to zero or infinitely far away. Yielding to this temptation, one can construct a coherent system, called the nonstandard reals, that includes infinitesimals and infinities, and thence proceed to do nonstandard analysis. Professor Loeb's research involves starting with problems in standard analysis, such as appear for instance in probability theory and mathematical physics, using nonstandard methods to gain some elbow room, and then getting back to a solution of the original problem in standard terms.