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In order to have the Lusin condition N (the property that sets of Lebesgue measure zero get mapped to sets of Lebesgue measure zero) for a Sobolev homeomorphism in the plane the assumption you need on the integrability of the differential is slightly less than power 2. Namely, in the plane it is sufficient to assume $$ \frac{|Df|^2}{\log(e+|Df|)} \in L_{\text{loc}}^1(U). $$

See [J. Kauhanen, P. Koskela and J. Malý, Mappings of nite finite distortion: condition N, Michigan Math. J. 49 (2001), no. 1, 169$-$181] http://projecteuclid.org/euclid.mmj/1008719040

There are change of variables formulas for Sobolev homeomorphisms. See for example [P. Hajĺasz, Change of variables formula under minimal assumptions, Colloq. Math. 64 (1993), 93$-$101] http://matwbn.icm.edu.pl/ksiazki/cm/cm64/cm64112.pdf and the references in it.
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In order to have the Lusin condition N (the property that sets of Lebesgue measure zero get mapped to sets of Lebesgue measure zero) for a Sobolev homeomorphism the assumption you need on the integrability of the differential is slightly less than power 2. Namely, in the plane it is sufficient to assume $$ \frac{|Df|^2}{\log(e+|Df|)} \in L_{\text{loc}}^1(U). $$

See [J. Kauhanen, P. Koskela and J. Malý, Mappings of nite distortion: condition N, Michigan Math. J. 49 (2001), no. 1, 169$-$181] http://projecteuclid.org/euclid.mmj/1008719040

There are change of variables formulas for Sobolev homeomorphisms. See for example [P. Hajĺasz, Change of variables formula under minimal assumptions, Colloq. Math. 64 (1993), 93$-$101] http://matwbn.icm.edu.pl/ksiazki/cm/cm64/cm64112.pdf and the references in it.