Using the variational properties of epi-convergence together with suitable results on the measurability of multifunctions and integrands, we prove a strong law of large numbers for sequences of integrands from which we deduce a general theorem of almost sure convergence (strong consistency) for the maximum likelihood estimator.
"Epi-convergence of sequences of normal integrands and strong consistency of the maximum likelihood estimator." Ann. Statist. 24 (3) 1298 - 1315, June 1996. https://doi.org/10.1214/aos/1032526970