Let $(X,\mathcal{A},\mu)$ be an atomless probability space and let $A\sim B$ whenever $\mu(A\vartriangle B)=0$ for each $A$ and $B$ in $\mathcal{A}$. This way, if $\mathbb{A}$ is the set of $\sim$-equivalences classes in $\mathcal{A}$, then $\mathbb{A}$ inherits $\cap,\cup,\cdot^c$ and $\mu$ from $(X,\mathcal{A},\mu)$, becoming a probability algebra. We can define a complete metric in $\mathbb{A}$ by $d([A],[B]):=\mu(A\vartriangle B)$, making it a structure in the sense of continuous model theory.
Given any measure preserving trasnformation $T:X\rightarrow X$, it defines a measure preserving $\sigma$-homomorphism $\check{T}:\mathbb{A}\rightarrow\mathbb{A}$ by $\check{T}([A]):=[T^{-1}(A)]$.
Is it true that for any measure preserving $\sigma$-homomorphism $\tau:\mathbb{A}\rightarrow\mathbb{A}$ there exists a measure preserving transformation $T:X\rightarrow X$ such that $\tau=\check{T}$? Is there a reference for such a theorem? If not, could you give a counterexample of such a $\tau$?
I know that this is true if $(X,\mathcal{A},\mu)$ is a Lebesgue-standard space (i.e. if $\mathbb{A}$ is separable), it is a theorem in Royden. But is it still true if $\mathbb{A}$ has a higher metric density?
It would be perfectly ok if this answer is restricted to just automorphisms.
Thanks, I appreciate any answer.