Eliminating positive-measure level sets by small Lipschitz perturbations

Abstract

We establish a new regularity phenomenon of continuous functions. Specifically, given any continuous function $f$ and arbitrary $\hat{I}\mu >0$, we construct a Lipschitz perturbation $g_{\hat{I}}\mu$ whose Lipschitz seminorm is less than $\hat{I}\mu$ such that every level set of $f+g_{\hat{I}}\mu$ has Lebesgue measure zero.