By Vitushkin A. G.

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That f WD uı is not absolutely continuous on Œ0; l . bj;i iD1 Let Ej D Snj 1 aj;i / < j 2 iD1 Œaj;i ; bj;i . aj;i /j Ä g ds ! 0; as j ! 1: jEj But this is a contradiction. e. curve. We shall next see that the equivalence classes for Newtonian functions are finer than for Sobolev functions. 19. 59. e. Proof. Without loss of generality we may assume that v Á 0 (consider otherwise C / D 0. We also u v). x/ ¤ 0g. e. curve. e. curve , u is both absolutely continuous and ƒ1 . E// D 0. e. with respect to arc length, and by continuity everywhere.

E. curve is such that gj is an upper gradient of fj along for all j D 1; 2; ::: , and neither nor any of its subcurves belong to . Consider such a curve W Œ0; l ! X . l / 2 E or Z Z jfQ. l // fQ. 0//j Ä lim sup jfj . l // fj . 51 shows that g is indeed a p-weak upper gradient of fQ. 49 shows that g is also a p-weak upper gradient of f . e. It will play the role of jruj in the rest of the book. Our main result here is the following theorem. 5. X /. e. e. X / of u. Moreover, gu is unique up to sets of measure zero.

44, we can assume that g1 and g2 are Borel functions. e. g1 Cg2 / ds < 1 and both g1 and g2 are upper gradients along . Let W Œ0; l ! 0; l / W g1 . t // Ä g2 . t //g. Un n E/ ! 0, as n ! 1, where ƒ1 is the one-dimensional S Lebesgue measure. 4. ) Then ju. 0// u. l //j Ä ju. 0// u. a1 //j C ju. a1 // C ju. b1 // u. l //j Z Z Ä g1 ds C g2 ds: jI1 u. b1 //j jI1 Continuing in this way, we obtain for all j D 1; 2; ::: , Z Z ju. 0// u. l //j Ä g1 ds C jSj I i D1 i jSj g2 ds: I i D1 i Letting j ! 1, monotone and dominated convergence show that Z Z ju.