Differentiation of sets in measure

Suppose F(ε)F(ε), for each ε∈[0,1]ε∈[0,1], is a bounded Borel subset of RdRd and F(ε)→F(0)F(ε)→F(0) as ε→0ε→0. Let A(ε)=F(ε)▵F(0)A(ε)=F(ε)▵F(0) be symmetric difference and PP be an absolutely continuous measure on RdRd. We introduce the notion of derivative of F(ε)F(ε) with respect to ε  , dF(ε)/dε=dA(ε)/dεdF(ε)/dε=dA(ε)/dε, such thatddεP(A(ε))|ε=0=Q(ddεA(ε)|ε=0), where QQ is another, explicitly described, measure, although not in RdRd.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 NnNn, n=1,2,…, is given on the class of set-valued mappings F={F(⋅)}F={F(⋅)} such that all F(ε)F(ε) converge to the same F=F(0)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.