Couplings, tightness, and existence

Work on Polish source and target spaces, meaning complete separable metric spaces, with probability marginals μ and ν. Each probability is tight: for any positive tolerance there is a compact set carrying all but that tolerance of its mass. Choose compact K and L for the two marginals. For every coupling, mass outside K × L is at most the source mass outside K plus the target mass outside L, by the union bound.

The same compact product works for every coupling because the marginals are fixed, giving uniform tightness of the feasible family. Prokhorov’s theorem then gives weakly convergent subsequences of any sequence of couplings. Compact support of the original distributions is unnecessary. On Euclidean space, finite moments can help estimate tails, but tightness of the fixed marginals already supplies the compactness step for a general nonnegative cost.

If couplings πₙ converge weakly to π, test the first coordinate with a bounded continuous function f(x). Its integral against every πₙ is the fixed integral of f against μ, so the same equality holds for π. Repeat for the second coordinate. The limiting measure is therefore still a coupling. This proves weak closure of the marginal constraints; weak convergence here is convergence against bounded continuous test functions.

Suppose the cost is nonnegative and lower semicontinuous on the product space. The Portmanteau inequality gives that the limit plan’s cost is at most the liminf of the sequence’s costs. Apply this to a minimizing sequence to obtain an optimizer. If at least one feasible coupling has finite cost, the optimum is finite. For squared Euclidean cost, finite second moments of both marginals make the independent coupling finite, using |x − y|² ≤ 2|x|² + 2|y|².

In a finite transport matrix, the feasible set is a closed bounded polytope and the linear objective attains its minimum. There can still be many minimizers, especially when some costs are equal. In continuous problems an optimal plan may likewise split mass or be singular. The direct method neither gives a unique optimizer nor produces a differentiable transport map. Those questions require extra geometry or strict structural assumptions, rather than more compactness.

Lower semicontinuity cannot be dropped casually. Raising a cost at the diagonal can make arbitrarily small shifts cheap while making their weak limit expensive. The practice problem turns this into a concrete nonattainment example on a compact square. This illustrates the roles of the assumptions cleanly: even compact feasible couplings cannot rescue an objective that jumps upward at its limiting candidate. Negative unbounded costs introduce further integrability issues, which are excluded by the nonnegative-cost theorem used here.