  For a given element $f\in L^1$ and a convex cone $C\subset L^\infty$, $C\cap
L^\infty_+=\{0\}$ we give necessary and sufficient conditions for the existence
of an element $g\ge f$ lying in the polar of $C$. This polar is taken in
$(L^\infty)^*$ and in $L^1$. In the context of mathematical finance the main
result concerns the existence of martingale measures, whose densities are
bounded from below by prescribed random variable.
