Let $(\Omega,\Sigma,\mu)$ be a countably generated probability space. Must $(\Omega,\Sigma,\mu)$ be isomorphic modulo null sets to a standard probability space?

I assume not, so here is a more specific question. Let $\Omega$ be an ultraproduct of finite sets and $\Sigma$ a countably generated sub-$\sigma$-algebra of the Loeb $\sigma$-algebra, and let $\mu$ be the Loeb measure. Is $(\Omega,\Sigma,\mu)$ a standard probability space?

I gather from Jin and Keisler. Maharam spectra of Loeb spaces, providing I understand the language, that if your ultraproducts are taken over a countable set in the usual way then the Loeb space is isomorphic modulo null sets to the product $\{0,1\}^\mathbf{R}$.

measure algebra isomorphismbetween the completion of $(\Omega,\Sigma,\mu)$ and a standard probability space. Do you mean a pointwise isomorphism up to a negligible set ? $\endgroup$ – Stéphane Laurent Mar 6 '16 at 11:42