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Download Cartesian Currents in the Calculus of Variations I: by Mariano Giaquinta PDF

(ii) cp E C,(X,R n) We say that vk converge to v in the mass norm , vk vk converge locally in the mass norm, vk + v, if MU(vk-v)--p0 if M(vk - v) -+ 0; VUCCX, U open.

5 Covering Theorems, Differentiation and Densities 31 We notice that Lemma 1 with X = R' is a consequence of Lemma 2. Later we shall refer to Lemma 1 and Lemma 2 as to Besicovitch's covering theorem. Let now µ and v be Radon measures over X. For x E X we set 17µv(x) := limsupµ(B(x,T)) r-O inf " a x,r Dµv(x) :_ lim r-O M(B(x,r)) whenever p(B(x, r)) > 0 for all r > 0, otherwise, i. , if A(B(x, r)) = 0 for some r > 0, we set Dµv(x) = Dµv(x) = +00. If Dµv(x) = Dµv(x) E [0, +oo], we simply write Dv(x) for Dµv(x) =Dµv(x).

I) Then there exists a continuous function g : IRn ---, R such that p{x E A I f (x) g(x)} < e. Notice that (iii) follows from (i) by means of Tietze's extension theorem for continuous functions on closed sets. Remark 1. Observing that RI is the union of a countable family of closed cubes with disjoint interiors and with side one, one easily sees that Lusin's theorem holds for any measurable set A of R', not necessarily of finite measure, if p is the Lebesgue measure C. This remark will be used later.

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