"The internal probability measures giving rise to a fixed Loeb measure"Schrittesser, DavidNonstandard analysis (invented by Abraham Robinson) makes the idea of infinitesimal and infinite real numbers precise using the ultrapower construction from model theory. A similar viewpoint can be taken in the construction of measure spaces: An "internal" measure (i.e., an ultrapower of measure spaces) gives rise to an "ordinary" measure, called its Loeb measure (as the construction is due to Peter Loeb). In fact, Lebesgue measure, or more generally, any Radon measure can be constructed in this manner from a hyperfinite internal measure (i.e., an ultraproduct of finite measure spaces). Although the Loeb construction has been a tool for many decades, even in its basic theory one quickly encounters hard and open questions (as is the case in the theory of "ordinary" measures, of course). Keisler and Sun (2004) asked several such questions, motivated by clarifying the interaction of the Loeb measure construction with the "ordinary" product measure construction. In recent joint work with William Weiss and Haosui Duanmu, all but one rather exotic case of these questions have been answered. In this talk, I will explain the Loeb construction, one of Keisler and Sun's questions, our solution, and the remaining open question. |
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