Multivariable Advanced Calculus

9.6. THE VITALI COVERING THEOREM 225Since the sets {W i } ∞ i=1 are locally finite, it follows ∪∞ i=1 W i = ∪ ∞ i=1 W i and so it ispossible to define ϕ i and γ, infinitely differentiable functions having compact supportsuch thatU i ≺ ϕ i ≺ V i , ∪ ∞ i=1W i ≺ γ ≺ ∪ ∞ i=1U i .Now define{ γ(x)ϕi (x)/ ∑ ∞j=1ψ i (x) =ϕ j(x) if ∑ ∞j=1 ϕ j(x) ≠ 0,∑ ∞0 ifj=1 ϕ j(x) = 0.If x is such that ∑ ∞j=1 ϕ j(x) = 0, then x /∈ ∪ ∞ i=1 U i because ϕ i equals one on U i .Consequently γ (y) = 0 for all y near x thanks to the fact that ∪ ∞ i=1 U i is closed andso∑ψ i (y) = 0 for all y near x. Hence ψ i is infinitely differentiable at such x. If∞j=1 ϕ j(x) ≠ 0, this situation persists near x because each ϕ j is continuous and so ψ iis infinitely differentiable at such points also. Therefore ψ i is infinitely differentiable. Ifx ∈ K, then γ (x) = 1 and so ∑ ∞j=1 ψ j(x) = 1. Clearly 0 ≤ ψ i (x) ≤ 1 and spt(ψ j ) ⊆ V j .This proves the theorem. The functions, {ψ i } are called a C ∞ partition of unity.The method of proof of this lemma easily implies the following useful corollary.Corollary 9.5.15 If H is a compact subset of V i for some V i there exists a partitionof unity such that ψ i (x) = 1 for all x ∈ H in addition to the conclusion of Lemma9.5.14.Proof: Keep V i the same but replace V j with Ṽj ≡ V j \ H. Now in the proof above,applied to this modified collection of open sets, if j ≠ i, ϕ j (x) = 0 whenever x ∈ H.Therefore, ψ i (x) = 1 on H. 9.6 The Vitali Covering TheoremThe Vitali covering theorem is a profound result about coverings of a set in R p withopen balls. The balls can be defined in terms of any norm for R p . For example, thenorm could be||x|| ≡ max {|x k | : k = 1, · · · , p}or the usual norm√ ∑|x| =k|x k | 2or any other. The proof given here is from Basic Analysis [27]. It first considers thecase of open balls and then generalizes to balls which may be neither open nor closed.Lemma 9.6.1 Let Fbe a countable collection of balls satisfying∞ > M ≡ sup{r : B(p, r) ∈ F} > 0and let k ∈ (0, ∞) . Then there exists G ⊆ F such thatIf B(p, r) ∈ G then r > k, (9.4)If B 1 , B 2 ∈ G then B 1 ∩ B 2 = ∅, (9.5)G is maximal with respect to 9.4 and 9.5. (9.6)By this is meant that if H is a collection of balls satisfying 9.4 and 9.5, then H cannotproperly contain G.