Is every sub-sigma-algebra of a countably generated sigma-algebra always isomorphic mod zero to a countably generated one?

Question

Let $(\Omega,\Sigma,p)$ be a probability space, and suppose that the $\sigma$-algebra $\Sigma$ is countably generated.

Let $\Sigma'\subseteq\Sigma$ be a sub-$\sigma$-algebra. Is it true that $(\Omega,\Sigma',p)$ is isomorphic mod zero to some standard probability space $(\Omega,\Phi,p)$, where $\Phi$ is countably generated?

I would also very much like a reference.