Loeb extension and Loeb equivalence II
Volume 262 / 2023
Abstract
The paper answers two open questions raised by Keisler and Sun (2004). 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 the work of Anderson et al. (2021), we give an affirmative 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 we 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.
