8.6. THE VITALI COVERING THEOREMS 199

Lemma 8.6.2 Let F be a collection of balls satisfying

∞ > M ≡ sup{r : B(p,r) ∈F}> 0

and let k ∈ (0,∞) . Then there exists G ⊆F such that

If B(p,r) ∈ G then r > k, (8.12)

If B1,B2 ∈ G then B1∩B2 = /0, (8.13)

G is maximal with respect to 8.12 and 8.13. (8.14)

By this is meant that if H is a collection of balls satisfying 8.12 and 8.13, then H cannotproperly contain G .

Proof: Let S denote a subset of F such that 8.12 and 8.13 are satisfied. Then ifS = /0, it means there is no ball having radius larger than k. Otherwise, S ̸= /0. Partiallyorder S with respect to set inclusion. Thus A ≺B for A ,B in S means that A ⊆B.By the Hausdorff maximal theorem, there is a maximal chain in S denoted by C . Then letG be ∪C . If B1,B2 are in C , then since C is a chain, both B1,B2 are in some element ofC and so B1∩B2 = /0. The maximality of C is violated if there is any other element of Swhich properly contains G . ■

Proposition 8.6.3 Let F be a collection of balls, and let A≡∪{B : B ∈F}. Suppose

∞ > M ≡ sup{r : B(p,r) ∈F}> 0.

Then there exists G ⊆F such that G consists of balls whose closures are disjoint and

A⊆ ∪{B̂ : B ∈ G }

where for B = B(x,r) a ball, B̂ denotes the open ball B(x,5r).

Proof: Let G1 satisfy 8.12, 8.13, 8.14 for k = 2M3 .

Suppose G1, · · · ,Gm−1 have been chosen for m ≥ 2. Let Gi denote the collection ofclosures of the balls of Gi. Then let Fm be those balls of F , such that if B is one of theseballs, B has empty intersection with every closed ball of Gi for each i≤ m−1. Then usingLemma 8.6.2, let Gm be a maximal collection of balls from Fm with the property that eachball has radius larger than

( 23

)mM and their closures are disjoint. Let G ≡ ∪∞

k=1Gk. Thusthe closures of balls in G are disjoint. Let x ∈ B(p,r) ∈F \G . Choose m such that(

23

)m

M < r ≤(

23

)m−1

M

Then B(p,r) must have nonempty intersection with the closure of some ball from G1∪·· ·∪Gm because if it didn’t, then Gm would fail to be maximal. Denote by B(p0,r0) a ball inG1∪ ·· ·∪Gm whose closure has nonempty intersection with B(p,r). Thus r0,r >

( 23

)mM.

Consider the picture, in which w ∈ B(p0,r0)∩B(p,r).

w••r0

p0•rp•x