Uniform approximation of continuous couplings (Q6629538)

Let \( X \) be the product space of probabilistic metric spaces \( (X_j,d_j,\mu_j), \) \( j=1,\ldots,N, \) \( N\geq 2 ,\) equipped with the product measure \( \mu \) and the distance which is equal to the sum of distances. A function \( f_j\in C(X_j) \) is a marginal, if its integral over \( X_j \) is equal to 1. Given \( \rho_1,\ldots,\rho_N \) marginals, the authors study the space of continuous couplings \N\[\N\Pi(\rho_1,\ldots,\rho_N)=\{\gamma\in C(X)|\gamma\geq 0|\pi_j\gamma=\rho_j \forall j=1,\ldots,N\}, \N\]\Nwhere \( \pi_j \) denotes the \( j\)-th marginal of \( \gamma, \) obtained by integrating \( \gamma\) with respect to all the reference measures \( \mu_1,\ldots,\mu_N \) except \( \mu_j. \)\N\NThe main result (Theorem 1.1): Suppose that spt \( \mu \) is compact. Let \( \gamma \) be a coupling, and \( \rho_j\in C (X_j) \) be marginals for \( j=1,\ldots,N. \) Then, for every \( \varepsilon >0 \) there exists \( \sigma (\varepsilon) >0 \) such that, if \( \Vert \pi_j\gamma - \rho_j\Vert_{L^{\infty}(\mu_j)}<\sigma (\varepsilon) \) for every \( j=1,\ldots,N \) then there exists \( \gamma'\in \Pi(\rho_1,\ldots,\rho_N) \) with \( \Vert \gamma -\gamma'\Vert_{L^{\infty}(\mu)<\varepsilon}. \)