A subcopula based dependence measure

a) 0 = FX,Y (−∞, y) = S(FX(−∞), FY (y)) = S(0, v) where v ∈ Ran FY , and analogously S(u, 0) = 0 for u ∈ Ran FX . b) FY (y) = FX,Y (+∞, y) = S(FX(+∞), FY (y)) = S(1, v) where v = FY (y), and analogously S(u, 1) = u where u = FX(x) for some x ∈ R. c) 0 ≤ P (x1 < X ≤ x2 , y1 < Y ≤ y2) = FX,Y (x2, y2)−FX,Y (x2, y1)−FX,Y (x1, y2)+ FX,Y (x1, y1) and therefore by (1) we have that S(u2, v2)− S(u2, v1)− S(u1, v2) + S(u1, v1) ≥ 0 where ui = FX(xi) and vi = FY (yi) for i = 1, 2.