Theorem 3  Given (P0) and (P1), (P7) and (P8) hold if and only if \(C_{P}(h, e\mid k)\) is a relative distance measure, that is, if there exists a strictly increasing function \(f\) such that, for any \(h, e, k\in \bL\) and any \(P\in \bP,\) \(C_{P}(h, e\mid k) = f[Z(h, e\mid k)],\) where: