In a model independent discrete time financial market, we discuss the richness of the family of martingale measures in relation to different notions of Arbitrage, generated by a class $mathcal{S}$ of significant sets, which we call Arbitrage de la classe $mathcal{S}$. The choice of $mathcal{S}$ reflects into the intrinsic properties of the class of polar sets of martingale measures. In particular: for S=${Omega}$ absence of Model Independent Arbitrage is equivalent to the existence of a martingale measure; for $mathcal{S}$ being the open sets, absence of Open Arbitrage is equivalent to the existence of full support martingale measures. These results are obtained by adopting a technical filtration enlargement and by constructing a universal aggregator of all arbitrage opportunities. We further introduce the notion of market feasibility and provide its characterization via arbitrage conditions. We conclude providing a dual representation of Open Arbitrage in terms of weakly open sets of probability measures, which highlights the robust nature of this concept.
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