Multivariable Advanced Calculus

226 THE LEBESGUE INTEGRAL FOR FUNCTIONS OF P VARIABLESProof: If no ball of F has radius larger than k, let G = ∅. Assume therefore, thatsome balls have radius larger than k. Let F ≡ {B i } ∞ i=1 . Now let B n 1be the first ballin the list which has radius greater than k. If every ball having radius larger than kintersects this one, then stop. The maximal set is {B n1 } . Otherwise, let B n2 be thenext ball having radius larger than k which is disjoint from B n1 . Continue this wayobtaining {B ni } ∞ i=1, a finite or infinite sequence of disjoint balls having radius largerthan k. Then let G ≡ {B ni }. To see G is maximal with respect to 9.4 and 9.5, supposeB ∈ F, B has radius larger than k, and G ∪ {B} satisfies 9.4 and 9.5. Then at somepoint in the process, B would have been chosen because it would be the ball of radiuslarger than k which has the smallest index. Therefore, B ∈ G and this shows G ismaximal with respect to 9.4 and 9.5. For an open ball, B = B (x, r) , denote by ˜B the open ball, B (x, 4r) .Lemma 9.6.2 Let Fbe a collection of open balls, and letSupposeA ≡ ∪ {B : B ∈ F} .∞ > M ≡ sup {r : B(p, r) ∈ F} > 0.Then there exists G ⊆ F such that G consists of disjoint balls andA ⊆ ∪{ ˜B : B ∈ G}.Proof: Without loss of generality assume F is countable. This is because there isa countable subset of F, F ′ such that ∪F ′ = A. To see this, consider the set of ballshaving rational radii and centers having all components rational. This is a countableset of balls and you should verify that every open set is the union of balls of this form.Therefore, you can consider the subset of this set of balls consisting of those which arecontained in some open set of F, G so ∪G = A and use the axiom of choice to define asubset of F consisting of a single set from F containing each set of G. Then this is F ′ .The union of these sets equals A . Then consider F ′ instead of F. Therefore, assumeat the outset F is countable.By Lemma 9.6.1, there exists G 1 ⊆ F which satisfies 9.4, 9.5, and 9.6 with k = 2M 3 .Suppose G 1 , · · · , G m−1 have been chosen for m ≥ 2. LetF m = {B ∈ F : B ⊆ R p \union of the balls in these G j{ }} {∪{G 1 ∪ · · · ∪ G m−1 } }and using Lemma 9.6.1, let G m be a maximal collection of disjoint balls from F m withthe property that each ball has radius larger than ( 2 m3)M. Let G ≡ ∪∞k=1G k . Letx ∈ B (p, r) ∈ F. Choose m such that) mM < r ≤) m−1M( 23( 23Then B (p, r) must have nonempty intersection with some ball from G 1 ∪· · ·∪G m becauseif it didn’t, then G m would fail to be maximal. Denote by B (p 0 , r 0 ) a ball in G 1 ∪· · ·∪G mwhich has nonempty intersection with B (p, r) . Thus( ) m 2r 0 > M.3Consider the picture, in which w ∈ B (p 0 , r 0 ) ∩ B (p, r) .