Multivariable Advanced Calculus

9.7. VITALI COVERINGS 227Then✛ r 0p 0w❄ rpx 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: For B one of these balls, say B (x, r) ⊇ B ⊇ B (x, r), denote by B 1 , theopen ball B ( )x, 5r4 . Let F1 ≡ {B 1 : B ∈ F} and let A 1 denote the union of the balls inF 1 . Apply Lemma 9.6.2 to F 1 to obtainA 1 ⊆ ∪{˜B 1 : B 1 ∈ G 1 }where G 1 consists of disjoint balls from F 1 . Now let G ≡ {B ∈ F : B 1 ∈ G 1 }. Thus Gconsists of disjoint balls from F because they are contained in the disjoint open balls,G 1 . ThenA ⊆ A 1 ⊆ ∪{˜B 1 : B 1 ∈ G 1 } = ∪{ ̂B : B ∈ G}because for B 1 = B ( x, 5r49.7 Vitali Coverings), it follows ˜B1 = B (x, 5r) = ̂B. This proves the theorem. There is another version of the Vitali covering theorem which is also of great importance.In this one, disjoint balls from the original set of balls almost cover the set, leaving outonly a set of measure zero. It is like packing a truck with stuff. You keep trying to fillin the holes with smaller and smaller things so as to not waste space. It is remarkablethat you can avoid wasting any space at all when you are dealing with balls of any sortprovided you can use arbitrarily small balls.