- journal article metadata
European Mathematical Society Publishing House
2017-12-25 23:40:02
Rendiconti del Seminario Matematico della Università di Padova
Rend. Sem. Mat. Univ. Padova
RSMUP
0041-8994
2240-2926
General
10.4171/RSMUP
http://www.ems-ph.org/doi/10.4171/RSMUP
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2013)
138
2017
0
Lusin type theorems for Radon measures
Andrea
Marchese
Universität Zürich, Switzerland
Lusin type approximation, Lipschitz function, porous set
We add to the literature the following observation. If $\mu$ is a singular measure on $\mathbb{R}^n$ which assigns measure zero to every porous set and $f\colon \mathbb{R}^n\rightarrow\mathbb{R}$ is a Lipschitz function which is non-differentiable $\mu$-a.e., then for every $C^1$ function $g\colon \mathbb{R}^n\rightarrow\mathbb{R}$ it holds $$\mu\{x\in\mathbb{R}^n\colon f(x)=g(x)\}=0.$$ 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.
Real functions
Approximations and expansions
193
207
10.4171/RSMUP/138-9
http://www.ems-ph.org/doi/10.4171/RSMUP/138-9
12
22
2017