Differentiation of sets in measure

Suppose F(ε), for each ε ∈ [0, 1], is a bounded Borel subset of Rd and F(ε)→F(0) as ε →0. Let A(ε) = F(ε) F(0) be symmetric difference and P be an absolutely continuous measure on Rd. We introduce the notion of derivative of F(ε) with respect to ε, dF(ε)/dε = dA(ε)/dε, such that d dε P A(ε) ε=0 = Q d dε A(ε) ε=0 , where Q is another, explicitly described, measure, although not in Rd . We discuss why this sort of derivative is needed to study local point processes in neighbourhood of a set: in short, if sequence of point processes Nn, n = 1, 2, . . . , is given on the class of set-valued mappings F = {F(·)} such that all F(ε) converge to the same F = F(0), then the weak limit of the local processes {Nn(A(ε)), F(ε) ∈ F} “lives” on the class of derivative sets {dF(ε)/dε|ε=0, F(·) ∈ F}. We compare this notion of the derivative set-valued mapping with other existing notions. © 2007 Elsevier Inc. All rights reserved.