We establish a quantitative version of the classical Halmos-Savage Theorem for convex, potentially non-dominated sets of probability measures and its dual counterpart, generalizing previous quantitative versions. These results are then used to derive robust versions of the fundamental theorem of asset pricing (FTAP) in large financial markets in a one-period setting, characterizing the absence of arbitrage under Knightian uncertainty. To this end, we consider robust formulations of no asymptotic arbitrage of first kind (NAA1), which is the large market analogue of ``No unbounded profit with bounded risk'' (NUPBR), as well as no asymptotic arbitrage of second kind (NAA2). Finally, we characterize asymptotic arbitrage of first and second kind in the robust one-period binomial model in terms of the model parameters.