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10

Part of 1974 Miklós Schweitzer

Problems(1)

Miklos Schweitzer 1974_10

Source:

11/12/2008
Let μ \mu and ν \nu be two probability measures on the Borel sets of the plane. Prove that there are random variables ξ1,ξ2,η1,η2 \xi_1, \xi_2, \eta_1, \eta_2 such that (a) the distribution of (ξ1,ξ2) (\xi_1, \xi_2) is μ \mu and the distribution of (η1,η2) (\eta_1, \eta_2) is ν \nu, (b) ξ1≤η1,ξ2≤η2 \xi_1 \leq \eta_1, \xi_2 \leq \eta_2 almost everywhere, if an only if μ(G)≥ν(G) \mu(G) \geq \nu(G) for all sets of the form G\equal{}\cup_{i\equal{}1}^k (\minus{}\infty, x_i) \times (\minus{}\infty, y_i). P. Major
probabilityprobability and stats