Before revision: It is well-known that an  ^n -valued random vector (X_1, X_2, \cdots, X_n) is comonotonic if and only if (X_1, X_2, \cdots, X_n) and (Q_1(U), Q_2(U),\cdots, Q_n(U)) coincide in distribution, for any random variable U uniformly distributed on the unit interval (0,1), where Q_k(\cdot) are the quantile functions of X_k, k=1,2,\cdots, n. It is natural to ask whether (X_1, X_2, \cdots, X_n) and (Q_1(U), Q_2(U),\cdots, Q_n(U)) can coincide almost surely for some special U.
After revision: It is well-known that an  \mathbb{R -valued random vector (X_1, X_2, \cdots, X_n) is comonotonic if and only if (X_1, X_2, \cdots, X_n) and (Q_1(U), Q_2(U),\cdots, Q_n(U)) coincide in distribution, for any random variable U uniformly distributed on the unit interval (0,1), where Q_k(\cdot) are the quantile functions of X_k, k=1,2,\cdots, n. It is natural to ask whether (X_1, X_2, \cdots, X_n) and (Q_1(U), Q_2(U),\cdots, Q_n(U)) can coincide almost surely for some special U.