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Let $\psi$ be a multi-dimensional random variable. We show that the set ofprobability measures $\mathbb{Q}$ such that the $\mathbb{Q}$-martingale$S^{\mathbb{Q}}_t=\mathbb{E}^{\mathbb{Q}}\left[\psi\lvert\mathcal{F}_{t}\right]$has the Martingale Representation Property (MRP) is either empty or dense in$\mathcal{L}_\infty$-norm. The proof is based on a related result involvinganalytic fields of terminal conditions $(\psi(x))_{x\in U}$ and probabilitymeasures $(\mathbb{Q}(x))_{x\in U}$ over an open set $U$. Namely, we show thatthe set of points $x\in U$ such that $S_t(x) =\mathbb{E}^{\mathbb{Q}(x)}\left[\psi(x)\lvert\mathcal{F}_{t}\right]$ does nothave the MRP, either coincides with $U$ or has Lebesgue measure zero. Our studyis motivated by the problem of endogenous completeness in financial economics.

Density of the set of probability measures with the martingale representation property