Sanov's statement of first-order asymptotic behaviour of probabilities of large deviations of an empirical distribution function is here established for empirical probability measures, with attendant simplification of conditions. For the case of distribution functions, our theorem is strictly more general than a specialisation of results of Hoadley.
"Large Deviations of Empirical Probability Measures." Ann. Statist. 2 (2) 362 - 366, March, 1974. https://doi.org/10.1214/aos/1176342671