Let:
- $\mathcal{M}$ be the $\sigma$-algebra induced on $[0, 1]$ by the Lebesgue measure $\mu$,
- $\mathbb{A}$ be the measure algebra $\mathcal{M} / \mathcal{N}$, i.e. the quotient of $\mathcal{M}$ by null sets $\mathcal{N}$,
- $\mathbb{B}$ be the subalgebra of $\mathbb{A}$ generated by a partition $[0, 1]$ into finitely many measurable sets with positive measure.
Is there a (measure preserving) isomorphism onto $\Phi: \mathbb{A} \rightarrow \mathbb{A}$ such that $\Phi(X) \not\in \mathbb{B}$ for some $X \in \mathbb{B}$?
Seems plausible that there is but am not sure how to (dis)prove it.