Transfer Operators from Batches of Unpaired Points via Entropic Transport Kernels

The paper’s Corollary 13 states exactly the Γ-convergence and minimizer stability result under compactness of Q, uniform positivity, and Γ-convergence of the indicators, and proves it by combining joint almost-sure convergence J_N^M(q_N)→J_M(q) along uniformly convergent sequences (Corollary 12) with the Γ-convergence of the indicator constraints and the fundamental theorem of Γ-convergence . The candidate solution follows the same structure, adding a Lipschitz argument and a countable-dense SLLN diagonalization to obtain an almost-sure event uniform over Q; the liminf/limsup steps and the minimizer conclusions match the paper’s proof. No substantive gaps relative to the paper’s assumptions were found.