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By Frank Morgan

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Let W be an mdimensional rectifiable subset of Rn , Z a -dimensional rectifiable subset of R , m ½ ½ 1, and f a Lipschitz function from W to Z. Then ap J fd H m D W Hm f 1 fzg d H z. Z More generally, for any H m W integrable function g on W, g Ð ap J fd H m D W g dH m z f 1 fzg d H z. Lipschitz Functions and Rectifiable Sets 31 Note: If f has an extension f to Rn , ap J f Ä J f (where both are defined). 14. 23]. Let W be an m-dimensional rectifiable Borel subset of Rn and let Z be a -dimensional rectifiable Borel subset of R .

For the rectifiable currents T 2 R1 R2 and for C1 functions f, g, and h, compute formulas for T f d x C g dy and ∂T h : (a) T D H 1 f x, 0 : 0 Ä x Ä 1g ^ e1 . p (b) T D H 1 f x, x : 0 Ä x Ä 1g ^ 3 2 e1 C e2 . 10. Prove that Im is M dense in Rm and F dense in Fm . 11. Prove that fT 2 Rm : spt T ² B 0, R g is M complete and that fT 2 Fm : spt T ² B 0, R g is F complete. 12. Prove that ∂ carries Nm into Nm 1 and Fm into Fm 1 . 13. 3(1): Nm D fT 2 Rm : M ∂T < 1g, Rm D fT 2 Fm : M T < 1g. 14. Prove that, in analogy with the definitions of Im and Fm , Nm D fT 2 Rm : ∂T 2 Rm 1 g, Fm D fT C ∂S: T 2 Rm , S 2 RmC1 g.

1. A purely unrectifiable one-dimensional set E. E is invisible from almost all directions. slope 12 ) gives an exceptional case: the projection is a solid interval. If A is any rectifiable set, then H 1 A \ E D 0. 1. Give an example of a Lipschitz function f: [0, 1] ! R such that f is not differentiable at any rational point. 2. 3 to deduce that a Lipschitz function is approximately differentiable almost everywhere. 3. Give an example of a continuous function f: R ! R such that (a) given ε > 0 there is a C1 function g: R1 !

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