Loeb Extension and Loeb Equivalence

TL;DR

This paper addresses open problems in Loeb equivalences between internal probability spaces by providing counter-examples and reducing a key problem to a new question about internal algebra generation.

Contribution

It offers counter-examples to initial open problems and reformulates a third problem in terms of internal algebra properties.

Findings

Counter-examples for the first two open problems.

Reduction of the third problem to an algebraic question.

New insights into Loeb equivalence and internal algebra structures.

Abstract

In Keisler and Sun (2004), the authors raise several open problems on Loeb equivalences between various internal probability spaces. We provide counter-examples for the first two open problems. Moreover, we reduce the third open problem to the following question: Is the internal algebra generated by the union of two Loeb equivalent internal algebras a subset of the Loeb extension of any one of the internal algebra?