SCHWEITZER 1974 Problem 10

Let $\mu$ and $\nu$ be two probability measures on the Borel sets of the plane. Prove that there are random variables $\xi_1, \xi_2, \eta_1, \eta_2$ such that

(a) the distribution of $(\xi_1, \xi_2)$ is $\mu$ and the distribution of $(\eta_1, \eta_2)$ is $\nu$ ,

(b) $\xi_1 \leq \eta_1, \xi_2 \leq \eta_2$ almost everywhere, if an only if $\mu(G) \geq \nu(G)$ for all sets of the form $G=\cup_{i=1}^k (-\infty, x_i) \times (-\infty, y_i).$

P. Major