Some fundamental properties of the empirical distribution function are derived in the case of independent but not necessarily identically distributed random variables. The distribution functions of these random variables need not be continuous.
"Properties of the Empirical Distribution Function for Independent Nonidentically Distributed Random Variables." Ann. Probab. 6 (2) 250 - 266, April, 1978. https://doi.org/10.1214/aop/1176995571