**"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. |