Professor Loeb is one of the world's leading experts in the applications of the fundamental foundational concept in mathematics called nonstandard analysis. This construct, first introduced by logicians, is an ingenious rigorous development of the classical concepts of infinitesimally large and small numbers. Nonstandard analysis, therefore, is a mathematical vindication of Leibnitz' philosophical viewpoint in his invention of the calculus. Professor Loeb is the leading practitioner in the use of nonstandard methods to prove "standard" results in a broad range of mathematical analysis and applications. Loeb's work in nonstandard measure theory and nonstandard measure theory and nonstandard probability theory has led to important contributions to probability theory, mathematical physics, and mathematical economics. The notion of Loeb measure, introduced by the principal investigator, has become a standard tool in probability theory, stochastic analysis, and potential theory. The current proposal by Professor Loeb has two parts. First, he proposes to combine his nonstandard integration theory with nonstandard convex hulls. Applications are envisaged to Banach spaces and Banach space valued stochastic processes. Secondly, he proposes to continue his work on the concept of ideal (Loeb) boundary, developed for very general potential theories. This research promises to have important implications for classical potential theory.
