No arbitrage of the first kind and local martingale numeraires

A supermartingale deflator (resp. local martingale deflator) multiplicatively transforms nonnegative wealth processes into supermartingales (resp. local martingales). A supermartingale numeraire (resp. local martingale numeraire) is a wealth process whose reciprocal is a supermartingale deflator (resp. local martingale deflator). It has been established in previous works that absence of arbitrage of the first kind (NA(1)) is equivalent to the existence of the (unique) supermartingale numeraire, and further equivalent to the existence of a strictly positive local martingale deflator; however, under NA(1), a local martingale numeraire may fail to exist. In this work, we establish that under NA(1), a supermartingale numeraire under the original probability P becomes a local martingale numeraire for equivalent probabilities arbitrarily close to P in the total variation distance.