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Sample text

Since by hypothesis U ∈ Σ(µ), we have that µ(A1 ∪ A2 ) = µ (A1 ∪ A2 ) ∩ U + µ (A1 ∪ A2 ) ∩ U c . , µ is metric outer measure. “⇐=”: It suffices to show that Σ(µ) contains all closed sets. So let C ⊆ X be df df closed and let us set U = C c . 2. Then d(An , C) © 2005 by Taylor & Francis Group, LLC 1 n ∀n 1 1 be an 1. 2, we have µ(D \ C) = µ(A) = lim µ(An ). 20) n→+∞ Since by hypothesis µ is a metric outer measure and the sets {An }n separated from C, we have µ(D) µ (D ∩ C) ∪ An = µ(D ∩ C) + µ(An ) ∀n 1 are 1.

C) The curve C is said to be rectifiable, if l(C) < +∞. , a compact and connected set in X. In particular then a curve is a Borel set; hence it is also µ(s) -measurable. 26(a) f is injective, then f −1 exists and is continuous and so C is the homeomorphic image of [0, 1]. 26(a), we can replace [0, 1] by any closed bounded interval [a, b]. Some authors require f to be injective. 28 If (X, d ) is a metric space, f : [0, 1] −→ X is a nonconstant curve with length l and C = f [0, 1] , then (a) 0 < µ(1) (C) l; (b) if f is injective, then µ(1) (C) = l.

T Since ε > 0 was arbitrary, we conclude that µ(s) (Ct ) = 0. 1, we need to check and see if something can be said about the density of A at its points. 4. a. x ∈ A. a. x ∈ A. 29) To this end, for every t > 1, we introduce the set Ct ⊆ A defined by df Ct = x ∈ A : lim sup r 0 µ(s) (B r (x) ∩ A) >t . (2r)s Fix ε > 0. 9). 10(b)). We introduce the family T of closed balls defined by df T = B r (x) : B r (x) ⊆ U, 0 < r < δ, µ(s) (B r (x) ∩ A) >t . 4, we can find a sequence B rn (xn ) joint balls in T , such that m n 1 of dis- ∞ Ct ⊆ B rn (xn ) ∪ n=1 B 5rn (xn ) ∀m 1.

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