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

勒布教授的项目涉及使用非标准 真实的分析和相关领域的模型。在他的基础上 方法是使用定义在 非标准点集这些空间使有限的组合 研究无限概率空间的方法。 他工作的另一个方面涉及非标准船体（标准， 由非标准向量空间形成的无限维向量空间） 向量格及其在测度论中的作用， 一体化他还将继续研究一个理想的边界， 类似于马丁边界，他已经开发了 一般势能理论 所有的数学分析都是建立在 由真实的数字组成的系统，这些数字被标识为 双向无限线这是常识和基本的， 这个系统，每一个非零的真实的数位于有限 距离为零，但有时 强大的智力诱惑，认为数字是无限的， 接近于零或无限远。屈服于此 诱惑，一个人可以构建一个连贯的系统，称为 非标准实数，包括无穷小和无穷大， 然后进行非标准分析。洛布教授 研究包括从标准分析中的问题开始， 例如出现在概率论中， 数学物理，使用非标准方法获得一些 然后回到原来的解 标准术语的问题。

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.