Since a density function f naturally lives on L₁, as does its estimator f_n,K based upon a kernel K, Devroye and Györfi have long advocated that the natural distance to measure the error in estimation between a density function f and f_n,K is the L₁-norm of their difference. We shall introduce a notion of an L₁-norm density estimator process indexed by a class of kernels, and then describe a functional central limit theorem and a Glivenko-Cantelli theorem that we obtained for this process. We shall also discuss some of the machinery that we developed to prove these results, which will likely be of separate interest. None of our theorems impose any condition at all on the underlying Lebesgue density f. Also, somewhat unexpectedly, the distribution of the limiting Gaussian process does not depend on f. This means that the process is asymptotically distribution free, which points the way to the potential use of our results to construct goodness-of-fit statistics. This is joint work with Evarist Giné and Andrei Zaitsev.