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A Lusin type approximation of Sobolev functions by smooth functions. (English) Zbl 0592.41031
Classical real analysis, Proc. Spec. Sess., 794th Meet. AMS, Madison/Wis. 1982, Contemp. Math. 42, 135-167 (1985).
[For the entire collection see Zbl 0565.00007.]
This paper is concerned with the approximation of a Sobolev function f by a smooth function g in such a way that $$| f-g|$$ is small and the set where f and g disagree has small capacity. The following is the main theorem. Let $$1\leq p<\infty$$, let $$\ell,m$$ be positive integers, with $$1\leq m\leq \ell$$ and $$(\ell -m)p<n$$ and let $$\Omega$$ be a non-empty open set of $$R^ n$$. Let $$f\in W^{\ell,p}(\Omega)$$ and be approximately continuous at each point of $$\Omega$$ except for a set E with Riesz capacity $$R_{\ell -m,p}(E)=0$$. Let $$\epsilon >0$$. Then there exists a $$C^ m$$ function g on $$\Omega$$, such that (a) the set $$F=\{x;x\in \Omega$$ and f(x)$$\neq g(x)\}$$ has $$R_{\ell -m,p}(F)<\epsilon$$ and (b) $$| f-g|_{m,h}<\epsilon.$$
In another theorem it is shown that each $$f\in W^{\ell,p}(\Omega)$$ can be represented by a function which is approximately continuous except for a set E, with $$R_{\ell -m,p}(E)=0$$. Since $$R_{0,p}$$ is equivalent to Lebesgue measure, the above theorem generalises a result of Fon-Che Liu [Indiana Univ. Math. J. 26, 645-651 (1977; Zbl 0368.46036)].

##### MSC:
 41A30 Approximation by other special function classes