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A METRIC ON PROBABILITIES, AND PRODUCTS OF LOEB SPACES

Published online by Cambridge University Press:  28 January 2004

H. JEROME KEISLER  and
YENENG SUN
H. JEROME KEISLER
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, WI 53706-1388, USAkeisler@math.wisc.edu
YENENG SUN
Affiliation:
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@nus.edu.sg
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Abstract

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.

Keywords

60B10 (primary)28E0560A10 (secondary)
Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 69 , Issue 1 , February 2004 , pp. 258 - 272
Copyright
The London Mathematical Society 2004

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Footnotes

This research was supported in part by the Vilas Trust Fund at the University of Wisconsin.