Compactness in the theory of large deviations

Large-deviation principles (LDPs) are expressed as the vague or narrow convergence of sequences of set functions called capacities. Compactness and other topological properties of the collection of capacities are then used in conjunction with Varadhanʹs integral theorem to reduce the proof of LDPs to the problem of showing that a certain system of equations has a unique solution. As applications of these ideas, we present short proofs of extended versions of a theorem of Bryc and of the Gنrtner-Ellis theorem.