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

Jp (X) where Jp (X) < oo. Using the lower semicontinuity of the total variation on open sets we have I p I (A) < 1iim of µk (A) I k-oo 38 1. General Measure Theory On the other hand for any closed set F we have lim SUP IIik I (F) = lim SUP (I Irk i (X) - Iuk I (X \ F) ) k-+oo k--+oo = I a K (X) - likm -oo f l /1k I (X \ F) >- l µ l (X) -III (X \ F) = I µ l (F) Thus by Theorem 3 we infer Proposition 1. , ILk IA,/LkI(X) --p III (X), Iul(X) < oo Then Ilk quasi-converges to (µ, WD- The next remark will be used later.

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