512 CHAPTER 19. HAUSDORFF SPACES AND MEASURES

Definition 19.5.2 If K is a compact subset of an open set V , then K ≺ φ ≺V if

φ ∈Cc(V ), φ(K) = {1}, φ(Ω)⊆ [0,1],

where Ω denotes the whole topological space considered. Also for φ ∈Cc(Ω), K ≺ φ if

φ(Ω)⊆ [0,1] and φ(K) = 1.

and φ ≺V if φ ∈Cc (V ) and

φ(Ω)⊆ [0,1] and spt(φ)⊆V.

Theorem 19.5.3 (Partition of unity) Let K be a compact subset of a locally compactHausdorff topological space satisfying Theorem 19.1.23 and suppose

K ⊆V = ∪ni=1Vi, Vi open.

Then there exist ψ i ≺Vi with ∑ni=1 ψ i(x) = 1 for all x ∈ K.

Proof: The proof is just like the one in Theorem 3.12.5 on Page 92. Let K1 =K\∪ni=2Vi.

Thus K1 is compact and K1 ⊆V1. Let K1 ⊆W1 ⊆W 1 ⊆V1 with W 1compact. To obtain W1,use Theorem 19.1.23 to get f such that K1 ≺ f ≺ V1 and let W1 ≡ {x : f (x) ̸= 0} . ThusW1,V2, · · ·Vn covers K and W 1 ⊆ V1. Let K2 = K \ (∪n

i=3Vi ∪W1). Then K2 is compactand K2 ⊆ V2. Let K2 ⊆W2 ⊆W 2 ⊆ V2, W 2 compact. Continue this way finally obtainingW1, · · · ,Wn, K ⊆W1 ∪ ·· · ∪Wn, and W i ⊆ Vi W i compact. Now let W i ⊆Ui ⊆U i ⊆ Vi ,U icompact.

Wi Ui Vi

By Theorem 19.1.23, let U i ≺ φ i ≺Vi, ∪ni=1W i ≺ γ ≺∪n

i=1Ui.Define

ψ i(x) ={

γ(x)φ i(x)/∑nj=1 φ j(x) if ∑

nj=1 φ j(x) ̸= 0,

0 if ∑nj=1 φ j(x) = 0.

If x is such that ∑nj=1 φ j(x) = 0, then x /∈ ∪n

i=1U i. Conse-quently γ(y) = 0 for all y near x and so ψ i(y) = 0 for all y near x. Hence ψ i is continuousat such x. If ∑

nj=1 φ j(x) ̸= 0, this situation persists near x and so ψ i is continuous at such

points. Therefore ψ i is continuous. If x ∈ K, then γ(x) = 1 and so ∑nj=1 ψ j(x) = 1. Clearly

0≤ ψ i (x)≤ 1 and spt(ψ j)⊆Vj. ■The following corollary won’t be needed immediately but is quite useful.

Corollary 19.5.4 In the context of the above theorem, if H is a compact subset of Vi,there exists a partition of unity such that ψ i (x) = 1 for all x ∈ H in addition to the conclu-sion of Theorem 19.5.3.

Proof: Keep Vi the same but replace Vj with Ṽj ≡ Vj \H. Now in the proof above, ap-plied to this modified collection of open sets, if j ̸= i,φ j (x) = 0 whenever x∈H. Therefore,ψ i (x) = 1 on H. ■

19.6 Measures on Hausdorff SpacesIn the case of a Hausdorff topological space, the following lemma gives conditions underwhich the σ algebra of µ measurable sets for an outer measure µ contains the Borel sets.