We show that given a monadically stable theory $T$, a sufficiently saturated $\mathbf M \models T$, and a coherent system of probability measures on the $σ$-algebras generated by parameter-definable sets of $\mathbf M$ in each dimension, we may produce a totally Borel $\mathbf B \prec \mathbf M$ realizing these measures. Our main application is to prove that every FO-convergent sequence of structures (with countable signature) from a monadically stable class admits a modeling limit. As another consequence, we prove a Borel removal lemma for monadically stable Lebesgue relational structures.