  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.
