19.5. PARTITIONS OF UNITY 511

Proof: Let ReA ≡ {Re f : f ∈A }, ImA ≡{Im f : f ∈A }.Claim 1: ReA = ImAProof of claim: A typical element of ReA is Re f where f ∈ A , then from Lemma

19.4.4, Re( f ) = Im(i f̄)∈ ImA . Thus ReA ⊆ ImA . By assumption, i f̄ ∈A . The other

direction works the same. Just use the other formula in Lemma 19.4.4.Claim 2: Both ReA and ImA are real algebras.Proof of claim: It is obvious these are both real vector spaces. Since these are equal, it

suffices to consider ReA . It remains to show that ReA is closed with respect to products.

f + f̄2

g+ ḡ2

=14[

f g+ f ḡ+ f̄ g+ f g]=

14[2Re( f g)+2Re

(f̄ g)]

Now by assumption, f g ∈A and so Re( f g) ∈ ReA . Also Re(

f̄ g)∈ ReA because both

f̄ ,g are in A and it is an algebra. Thus, the above is in ReA because, as noted, this is areal vector space.

Claim 3: A = ReA + i ImAProof of claim: If f ∈ A , then f = f+ f̄

2 + i f− f̄2i ∈ ReA + i ImA so A ⊆ ReA +

i ImA . Now a generic element of ReA + i ImA is Re( f )+ i Im(g) for f ,g ∈A .

Re( f )+ i Im(g)≡ f + f̄2

+ i(

g− ḡ2i

)=

f +g2

+f̄ − ḡ

2∈A

because A is closed with respect to conjugates. Thus ReA + i ImA ⊆A .Both ReA and ImA must separate the points. Here is why: If x1 ̸= x2, then there exists

f ∈A such that f (x1) ̸= f (x2) . If Im f (x1) ̸= Im f (x2) , this shows there is a function inImA , Im f which separates these two points. If Im f fails to separate the two points, thenRe f must separate the points and so, by Lemma 19.4.4,

Re f (x1) = Im(i f̄ (x1)

)̸= Re f (x2) = Im

(i f̄ (x2)

)Thus ImA separages the points. Similarly ReA separates the points using a similar argu-ment or because it is equal to ImA .

Neither ReA nor ImA annihilate any point. This is easy to see because if x is apoint, there exists f ∈ A such that f (x) ̸= 0. Thus either Re f (x) ̸= 0 or Im f (x) ̸= 0. IfIm f (x) ̸= 0, this shows this point is not annihilated by ImA . Since they are equal, ReAdoes not annihilate this point either.

It follows from Theorem 19.4.3 that ReA and ImA are dense in the real valued func-tions of C0 (X). Let f ∈C0 (X) . Then there exists {hn} ⊆ReA and {gn} ⊆ ImA such thathn→ Re f uniformly and gn→ Im f uniformly. Therefore, hn + ign ∈A and it convergesto f uniformly. ■

19.5 Partitions of UnityAs before, the idea of a partition of unity if of fundamental significance. It will be used toconstruct measures.

Definition 19.5.1 Define spt( f ) (support of f ) to be the closure of the set {x :f (x) ̸= 0}. If V is an open set, Cc(V ) will be the set of continuous functions f , definedon Ω having spt( f )⊆V . Thus in Theorem 19.1.23, f ∈Cc(V ).