Loeb extension and Loeb equivalence II

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.