Loeb Extension and Loeb Equivalence II

TL;DR

This paper addresses open questions about Loeb equivalence and internal probability measures, providing positive results for hyperfinite spaces, partial answers generally, and a counter-example for the $\sigma$-product versus product spaces.

Contribution

It resolves two open problems in Loeb measure theory, including conditions for Loeb equivalence and the nature of $\sigma$-products, advancing understanding of internal probability spaces.

Findings

Confirmed Loeb equivalence for hyperfinite internal spaces

Provided a counter-example for $\sigma$-product and product spaces

Showed continuity sets are in the algebra of the product space

Abstract

The paper answers two open questions that were raised in by Keisler and Sun. The first question asks, if we have two Loeb equivalent spaces $(\Omega, \mathcal F, \mu)$ and $(\Omega, \mathcal G, \nu)$, does there exist an internal probability measure $P$ defined on the internal algebra $\mathcal H$ generated from $\mathcal F\cup \mathcal G$ such that $(\Omega, \mathcal H, P)$ is Loeb equivalent to $(\Omega, \mathcal F, \mu)$? The second open problem asks if the $\sigma$-product of two $\sigma$-additive probability spaces is Loeb equivalent to the product of the same two $\sigma$-additive probability spaces. Continuing work in a previous paper, we give a confirmative answer to the first problem when the underlying internal probability spaces are hyperfinite, a partial answer to the first problem for general internal probability spaces, and settle the second question negatively by giving a counter-example. Finally, we show that the continuity sets in the $\sigma$-algebra of the $\sigma$-product space are also in the algebra of the product space.