Multivariable Advanced Calculus

422 HAUSDORFF MEASURES AND AREA FORMULAThusH s δ(A ∪ B˙) + ε > ∑ j∈J 1β(s)(r (C j )) s + ∑ j∈J 2β(s)(r (C j )) swhereJ 1 = {j : C j ∩ A ≠ ∅}, J 2 = {j : C j ∩ B ≠ ∅}.Recall dist(A, B) = 2δ 0 and so J 1 ∩ J 2 = ∅. It followsH s δ(A ∪ B) + ε > H s δ(A) + H s δ(B).Letting δ → 0, and noting ε > 0 was arbitrary, yieldsH s (A ∪ B) ≥ H s (A) + H s (B).Equality holds because H s is an outer measure. By Caratheodory’s criterion, H s is aBorel measure.To verify the second assertion, note first there is no loss of generality in lettingH s (E) < ∞. LetE ⊆ ∪ ∞ j=1C j , r(C j ) < δ,andLetThus F δ ⊇ E andH s δ(E) + δ >∞∑β(s)(r (C j )) s .j=1F δ = ∪ ∞ j=1C j .H s δ(E) ≤ H s δ(F δ ) ≤=∞∑β(s)(r ( )C j )sj=1∞∑β(s)(r (C j )) s < δ + Hδ(E).sj=1Let δ k → 0 and let F = ∩ ∞ k=1 F δ k. Then F ⊇ E andLetting k → ∞,H s δ k(E) ≤ H s δ k(F ) ≤ H s δ k(F δ ) ≤ δ k + H s δ k(E).H s (E) ≤ H s (F ) ≤ H s (E)This proves the theorem. A measure satisfying the first conclusion of Theorem 15.1.5 is sometimes called aBorel regular measure.15.1.2 H n And m nNext I will compare H n and m n . To do this, recall the following covering theorem whichis a summary of Corollaries 9.7.5 and 9.7.4 found on Page 230.Theorem 15.1.6 Let E ⊆ R n and let F, be a collection of balls of bounded radiisuch that F covers E in the sense of Vitali. Then there exists a countable collection ofdisjoint balls from F, {B j } ∞ j=1 , such that m n(E \ ∪ ∞ j=1 B j) = 0.In the next lemma, the balls are the usual balls taken with respect to the usualdistance in R n .