Stability of sigma-martingale densities in L log L under an equivalent change of measure

An equivalent sigma-martingale measure (EsigmaMM) for a given stochastic process S is a probability measure R equivalent to the original measure P such that S is an R-sigma-martingale. Existence of an EsigmaMM is equivalent to a classical absence-of-arbitrage property of S, and is invariant if we replace the reference measure P with an equivalent measure Q. Now suppose that there exists an E!MM R for S such that the density dR dP is in L logL(P). Does this existence property also remain invariant if we replace P by some equivalent Q? We prove that the answer is Yes if one imposes instead of a global only a local integrability requirement. sigma-martingale ; equivalent martingale measures ; Jacod decomposition ; mathematical finance