190 CHAPTER 8. POSITIVE LINEAR FUNCTIONALS

Taking sup for all such h, it follows that Lg ≥ (1−α)µ (Vα) ≥ (1−α)µ (K). Lettingα → 0 yields Lg ≥ µ(K). This proves the first part of the lemma. The second assertionfollows from this and Theorem 8.1.3. If K is given, let K ≺ g ≺ X and so from what wasjust shown, µ (K)≤ Lg < ∞. ■

For two sets A,B, recall dist(A,B)≡ inf{|a−b| : a ∈ A,b ∈ B} .

Lemma 8.2.5 If A and B are disjoint subsets of X, with dist(A,B)> 0 then µ(A∪B) =µ(A)+µ(B).

Proof: There is nothing to show if µ (A∪B) = ∞ so assume µ (A∪B) < ∞. Let δ ≡dist(A,B)> 0. Then let U1 ≡ ∪a∈AB

(a, δ

3

),V1 ≡ ∪b∈BB

(b, δ

3

). It follows that these two

open sets have empty intersection. Also, there exists W ⊇ A∪B such that µ (W )− ε <µ (A∪B). let U ≡U1∩W,V ≡V1∩W. Then

µ (A∪B)+ ε > µ (W )≥ µ (U ∪V )

Now let f ≺U,g≺V such that L f + ε > µ (U) ,Lg+ ε > µ (V ) . Then

µ (U ∪V ) ≥ L( f +g) = L( f )+L(g)

> µ (U)− ε +(µ (V )− ε)

≥ µ (A)+µ (B)−2ε

It follows thatµ (A∪B)+ ε > µ (A)+µ (B)−2ε

and since ε is arbitrary, µ (A∪B)≥ µ (A)+µ (B)≥ µ (A∪B). ■It follows from Theorem 6.7.2 that the σ algebra of measurable sets F determined

by this outer measure µ contains the Borel σ algebra B (X). Since closures of balls arecompact, it follows from Lemma 8.2.4 that µ is finite on every ball.

From the definition, for any E ∈F ,

µ (E) = inf{µ (V ) : V ⊇ E,V open}

Lemma 8.2.6 If µ is outer regular and F is measurable and contained in a closed ballB, then

µ (F) = sup{µ (K) : K ⊆ F,K compact}

Proof: By outer regularity, there exists V open with V ⊇ B∩FC and

µ (V \ (B\F))< ε.

Thus VC ⊆ BC ∪F and VC ∩B⊆(BC ∪F

)∩B = B∩F.

VC∩B

B\F

F