Lusin type theorems for Radon measures

  • Andrea Marchese

    Universität Zürich, Switzerland


We add to the literature the following observation. If μ\mu is a singular measure on Rn\mathbb{R}^n which assigns measure zero to every porous set and f ⁣:RnRf\colon \mathbb{R}^n\rightarrow\mathbb{R} is a Lipschitz function which is non-differentiable μ\mu-a.e., then for every C1C^1 function g ⁣:RnRg\colon \mathbb{R}^n\rightarrow\mathbb{R} it holds

μ{xRn ⁣:f(x)=g(x)}=0.\mu\{x\in\mathbb{R}^n\colon f(x)=g(x)\}=0.

In other words the Lusin type approximation property of Lipschitz functions with C1C^1 functions does not hold with respect to a general Radon measure.

Cite this article

Andrea Marchese, Lusin type theorems for Radon measures. Rend. Sem. Mat. Univ. Padova 138 (2017), pp. 193–207

DOI 10.4171/RSMUP/138-9