Lusin type theorems for Radon measures

Marchese, Andrea

Rendiconti del Seminario Matematico della Università di Padova, Tome 138 (2017), pp. 193-207.

We add to the literature the following observation. If $\mu$ is a singular measure on ${ℝ}^n$ which assigns measure zero to every porous set and $f:{ℝ}^n\to ℝ$ is a Lipschitz function which is non-differentiable $\mu$-a.e., then for every $C^1$ function $g:{ℝ}^n\to ℝ$ it holds

$$\mu \{x\in {ℝ}^n:f\left(x\right)=g\left(x\right)\}=0.$$

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

Classification : 26, 41
Mots-clés : Lusin type approximation, Lipschitz function, porous set