Differentiation of sets in measure

Abstract

Suppose F ( ε ) , for each ε ∈ [ 0 , 1 ] , is a bounded Borel subset of R d 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 R d . We introduce the notion of derivative of F ( ε ) with respect to ε, d F ( ε ) / d ε = d A ( ε ) / d ε , such that d d ε P ( A ( ε ) ) | ε = 0 = Q ( d d ε A ( ε ) | ε = 0 ) , where Q is another, explicitly described, measure, although not in R d . 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 N n , 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 { N n ( A ( ε ) ) , F ( ε ) ∈ F } “lives” on the class of derivative sets { d F ( ε ) / d ε | ε = 0 , F ( ⋅ ) ∈ F } . We compare this notion of the derivative set-valued mapping with other existing notions.