A Note on Convergence of Level Sets

Abstract

Given a sequence of functions $f_n$ converging in some topology to a function $f$, in general the 0-level set of $f_n$ does not give a good approximation of the one of $g$. In this paper we show that, if we consider an appropriate perturbation of the 0-level set of $f_n$, we get a sequence of sets converging to the 0-level set of $f$, where the type of set convergence depends on the type of convergence of $f_n$ to $f$.