Chebyshev inequalities and comonotonicity (Q1322640)

Two functions \(f\) and \(g\) on a set \(\Omega\) are comonotonic if \(f(x) > f(y)\) and \(g(y) > g(x)\) is impossible for any \(\{x,y\} \subset \Omega\). The author proves that, given a measurable space \((\Omega,\Sigma)\), two \(\Sigma\)-measurable real functions \(f\) and \(g\) on \(\Omega\) satisfy \(\int fdP \int gdP \leq \int fgdP\) for all countably additive probabilities \(P\) if and only if they are comonotonic.