  We define the local empirical process, based on $n$ i.i.d. random vectors in
dimension $d$, in the neighborhood of the boundary of a fixed set. Under
natural conditions on the shrinking neighborhood, we show that, for these local
empirical processes, indexed by classes of sets that vary with $n$ and satisfy
certain conditions, an appropriately defined uniform central limit theorem
holds. The concept of differentiation of sets in measure is very convenient for
developing the results. Some examples and statistical applications are also
presented.
