  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.
