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

Let . be a multidimensional random variable. We show that the set of probability measures Q such that the Q-martingale SQt=EQ[.|Ft] has the Martingale Representation Property (MRP) is either empty or dense in L.-norm. The proof is based on a related result involving analytic fields of terminal conditions (.(x))x.U and probability measures (Q(x))x.U over an open set U. Namely, we show that the set of points x.U such that St(x)=EQ(x)[.(x)|Ft] does not have the MRP, either coincides with U or has Lebesgue measure zero. Our study is motivated by the problem of endogenous completeness in financial economics.