We consider a set Q of probability measures, which are absolutely continuous with respect to the physical probability measure P and at least one is equivalent to P. We investigate in the finite probability space case, necessary and sufficient conditions on Q, under which any Q-supermartingale can be decomposed into the sum of a Q-martingale and a decreasing process. We also provide an orthogonal decomposition of Q-super-martingales and apply this result to the orthogonal decomposition of the polar cone of Q.