When we assume a superpopulation model in finite population, it is important to focus on the distribution function of the character of interest generated by the superpopulation. In this work we start studying a variation of the classical non-parametric empirical process. It is shown that the Hajék estimator of the population distribution function properly scaled and centered, converges weakly to a Gaussian process. The limiting process can be represented as the sum of two independent Gaussian processes. One of them is proportional to a Brownian bridge, while the other possesses a complex covariance kernel. Under some additional assumptions both of the process are proportional to a Brownian bridge. In applications, usually attention is focused on a functional of the superpopulation distribution function. In order to study this functional form, the Hadamard-derivative definition is introduced and it is shown that Hadamard differentiable functionals have some good properties with reference to weak convergence. To put into practice the theory a resampling scheme is introduced and it is shown that the resampled process converges to the same limit of the original considered process. At the end several applications for inference about superpopulation's quantities of interest will be presented, followed by Monte Carlo simulations.