On pairs of sums of random variables with painwise equal distributions

Let X(1), X(2), Y-1, Y-2 be four real-valued random variables such that X(1), X(2) have a common distribution function F, and Y-1 and Y-2 have a common distribution function G. Furstenberg (1967) asked: If X(1) + Y-1 greater than or equal to X(2) + Y-2 (1) holds, does it follow that equality holds with probability 1? Obviously, the answer is positive if the expectations of X(1) and Y-1 exist. We show that the existence of the expectation of X(1) is sufficient. We also show by example (announced by Krengel (1970)) that the answer is negative if both X(1) and Y-1 are allowed to have infinite expectation, even if all random variables are nonnegative.