Let $A$ be a definable subset of $\mathbb{R}$ in $\mathsf{ZF}$, and let $\mathcal{M},\mathcal{N}\models\mathsf{ZF}$ such that $A$ is lebesgue measurable in both models.

Is $\mu^\mathcal{M}(A^\mathcal{M})=\mu^\mathcal{N}(A^\mathcal{N})$?

Are there any conditions (applied to $A$ or $\mathcal{M},\mathcal{N}$) under which we know more about this question?

term relations. You can read about this in the Feng-Magidor-Woodin paper on universally Bare sets, and in Steel's first paper on the derived model theorem. This is now part of the core model induction machinery. $\endgroup$ – Andrés E. Caicedo Aug 4 '14 at 16:38