 Compact-parameter consistency of maximum likelihood

On a compact parameter space, suppose the expected log-likelihood is continuous, uniquely maximized at the true parameter, and the normalized empirical log-likelihood converges uniformly in probability to its expectation. On the compact complement of every neighbourhood of the truth, the expected criterion has a strictly positive gap below its maximum. Uniform approximation by less than one-third of this gap excludes every empirical maximizer from that complement. Identifiability and nonnegativity of Kullback-Leibler divergence supply the unique expected maximizer. The proof works for any measurable choice of maximizer.