518 CHAPTER 19. HAUSDORFF SPACES AND MEASURES

From Lemma 19.7.5 µ(A∪B)< ∞ and so there exists an open set, W such that

W ⊇ A∪B, µ (A∪B)+ ε > µ (W ) .

Now let U =U1∩W and V =V1∩W . Then

U ⊇ A, V ⊇ B, U ∩V = /0,and µ(A∪B)+ ε ≥ µ (W )≥ µ(U ∪V ).

Let A≺ f ≺U, B≺ g≺V . Then by Lemma 19.7.5,

µ(A∪B)+ ε ≥ µ(U ∪V )≥ L( f +g) = L f +Lg≥ µ(A)+µ(B).

Since ε > 0 is arbitrary, this proves the lemma. ■From Lemma 19.7.5 the following lemma is obtained.

Lemma 19.7.7 Let f ∈Cc(Ω), f (Ω) ⊆ [0,1]. Then µ(spt( f )) ≥ L f . Also, every openset, V satisfies µ (V ) = sup{µ (K) : K ⊆V} .

Proof: Let V ⊇ spt( f ) and let spt( f ) ≺ g ≺ V . Then L f ≤ Lg ≤ µ(V ) because f ≤ g.Since this holds for all V ⊇ spt( f ), L f ≤ µ(spt( f )) by definition of µ .

Vspt( f ) spt(g)

Finally, let V be open and let l < µ (V ) . Then from the definition of µ, there existsf ≺ V such that L( f ) > l. Therefore, l < µ (spt( f )) ≤ µ (V ) and so this shows the claimabout inner regularity of the measure on an open set. ■

At this point, the conditions of Lemma 19.6.1 have been verified. Thus S contains theBorel sets and µ is inner regular on sets of S having finite measure.

It remains to show µ satisfies 19.9.

Lemma 19.7.8 ∫f dµ = L f for all f ∈Cc(Ω).

Proof: Let f ∈Cc(Ω), f real-valued, and suppose f (Ω)⊆ [a,b]. Choose t0 < a and lett0 < t1 < · · ·< tn = b, ti− ti−1 < ε . Let

Ei = f−1((ti−1, ti])∩ spt( f ). (19.10)

Note that ∪ni=1Ei = spt( f ) since Ω = ∪n

i=1 f−1((ti−1, ti]). Let Vi ⊇ Ei,Vi is open and let Visatisfy

f (x)< ti + ε for all x ∈Vi, µ(Vi \Ei)< ε/n. (19.11)

By Theorem 19.5.3 there exists hi ∈ Cc(Ω) such that hi ≺ Vi, ∑ni=1 hi(x) = 1 on spt( f ).

Now note that for each i, f (x)hi(x) ≤ hi(x)(ti + ε). (If x ∈ Vi, this follows from 19.11. Ifx /∈Vi both sides equal 0.) Therefore,

L f = L(n

∑i=1

f hi)≤ L(n

∑i=1

hi(ti + ε)) =n

∑i=1

(ti + ε)L(hi)

=n

∑i=1

(|t0|+ ti + ε)L(hi)−|t0|L

=1 on spt( f )(n

∑i=1

hi

).