200 CHAPTER 8. POSITIVE LINEAR FUNCTIONALS

Then for x ∈ B(p,r),

|∥x−p0∥ ≤ ∥x−p∥+∥p−w∥+

≤r0︷ ︸︸ ︷∥w−p0∥

≤ r+ r+ r0 ≤ 2

< 32 r0︷ ︸︸ ︷(

23

)m−1

M+ r0 ≤ 2(

32

) <r0︷ ︸︸ ︷(23

)m

M+ r0 ≤ 4r0

Thus B(p,r) is contained in B(p0,4r0). It follows that the closures of the balls of G aredisjoint and the set

{B̂ : B ∈ G

}covers A. ■

Next is a version of the Vitali covering theorem which involves covering with disjointclosed balls. Here is the concept of a Vitali covering.

Definition 8.6.4 Let S be a set and let C be a covering of S meaning that everypoint of S is contained in a set of C . This covering is said to be a Vitali covering if for eachε > 0 and x ∈ S, there exists a set B ∈ C containing x, the diameter of B is less than ε, andthere exists an upper bound to the set of diameters of sets of C .

The following corollary is a consequence of the above Vitali covering theorem.

Corollary 8.6.5 Let F be a bounded set and let C be a Vitali covering of F consistingof closed balls. Let r (B) denote the radius of one of these balls. Then assume also thatsup{r (B) : B ∈ C }= M < ∞. Then there is a countable subset of C denoted by {Bi} suchthat m̄p

(F \∪N

i=1Bi)= 0 for N ≤ ∞, and Bi∩B j = /0 whenever i ̸= j.

Proof: Let U be a bounded open set containing F such that U approximates F so wellthat

mp (U)≤ rm̄p (F) ,r > 1 and very close to 1, r−5−p ≡ θ̂ p < 1

Since this is a Vitali covering, for each x∈ F, there is one of these balls B containing x suchthat B̂⊆U . Let Ĉ denote those balls of C such that B̂⊆U also. Thus, this is also a coverof F . By the Vitali covering theorem above, there are disjoint balls from C , {Bi} such that{

B̂i}

covers F . Thus

m̄p(F \∪∞

j=1B j)≤ mp

(U \∪∞

j=1B j)= mp (U)−

∞

∑j=1

mp (B j)

≤ rm̄p (F)−5−p∞

∑j=1

mp

(B̂ j

)≤ rm̄p (F)−5−pm̄p (F)

≡(r−5−p) m̄p (F)≡ θ̂ pm̄p (F)

Now if n1 is large enough and θ p is chosen such that 1 > θ p > θ̂ p, then

m̄p

(F \∪n1

j=1B j

)≤ mp

(U \∪n1

j=1B j

)≤ θ pm̄p (F) .

If m̄(

F \∪n1j=1B j

)= 0, stop. Otherwise, do for F \∪n1

j=1B j exactly the same thing that

was done for F. Since ∪n1j=1B j is closed, you can arrange to have the approximating open