A Local Property of Measurable Sets

Publisher Website
Google Scholar
Abstract
Let Ω be a metric space with metric ρ, let C be a class of closed sets from Ω and let τ be a non-negative real-valued set function on C. We assume that the empty set ϕ is in C and that τ(I)= 0 if and only if I = ϕ. For each set A in Ω, we define φ(A), 0 ≤ φ(A) ≤ ∞ by: where the infimum is taken for all possible countable collections of sets I(n) from C such that: and the diameter of I(n), d(I(n)), is less than ∈ for every n.