© 2004 by London Mathematical Society
© The London Mathematical Society
A Metric on Probabilities, and Products of Loeb Spaces
Department of Mathematics, University of Wisconsin-Madison 480 Lincoln Drive, Madison, WI 53706-1388, USA, keisler{at}math.wisc.edu
Institute for Mathematical Sciences, National University of Singapore 3 Prince George's Park, Singapore 118402
Department of Mathematics, National University of Singapore 2 Science Drive 2, Singapore 117543, matsuny{at}nus.edu.sg
Received 2 July 2002. Revision received 13 January 2003.
Two functions on finitely additive probability spaces that behave well under products are introduced: discrepancy, which measures how close one space comes to extending another, and bi-discrepancy, which is a pseudo-metric on the collection of all spaces on a given set, and a metric on the collection of complete spaces. These are then applied to show that the Loeb space of the internal product of two internal finitely additive probability spaces depends only on the Loeb spaces of the two original internal spaces. Thus the notion of a Loeb product of two Loeb spaces is well defined. The Loeb operation induces an isometry from the nonstandard hull of the space of internal probability spaces on a given set to the space of Loeb spaces on that set, with the metric of bi-discrepancy.