A Denoised Version of Some Kinds of Set Convergence
Filomena A. Lops · Advanced Nonlinear Studies · 2005
Abstract The aim of this paper consists of introducing on a locally compact and σ-compact metric space a notion of set convergence, which generalizes the Hausdorff convergence, the local Hausdorff convergence and the Kuratowski convergence. We analyze the connections beetwen the three new notions: and. in particular, we prove a compactness result. As a first application of this convergence we give, on a sequence of sets, a condition which assures the lower semicontinuity of the Hausdorff measure with respect to this new convergence and we show that this condition is satisfied by any minimizing sequence of Mumford-Shah functional.