8.2. POSITIVE LINEAR FUNCTIONALS AND MEASURES 191

In the picture B \F is pink between the solid circle and the solid ellipse and F is ingreen. The open set V is between the two dashed lines.

Then B∩VC is a compact subset of F and

µ(F \(B∩VC))≤ µ (V \ (B\F))< ε

and this shows that µ (F)− ε ≤ µ(B∩VC

)≤ µ (F) and so µ is inner regular for F . ■

If F is not necessarily contained in a closed ball, let Bn be a sequence of closed ballshaving increasing radii and let Fn = Bn ∩F . Then if l < µ (F) , it follows that µ (Fn) > lfor all large enough n. Then picking one of these, it follows from what was just shown thatthere is a compact set K ⊆ Fn such that also µ (K)> l.

Thus F contains the Borel sets and µ is inner regular on all sets of F , outer regular bydefinition.

It remains to show µ satisfies 8.6.

Lemma 8.2.7 ∫f dµ = L f for all f ∈Cc(X).

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

Ei = f−1((ti−1, ti])∩ spt( f ). (8.7)

Note that ∪ni=1Ei is a closed set equal to spt( f ).

∪ni=1Ei = spt( f ) (8.8)

Since X = ∪ni=1 f−1((ti−1, ti]). Let Vi ⊇ Ei,Vi is open and let Vi satisfy

f (x)< ti + ε for all x ∈Vi, µ(Vi \Ei)< ε/n. (8.9)

By Theorem 8.1.4, there exists hi ∈Cc(X) such that

hi ≺Vi,n

∑i=1

hi(x) = 1 on spt( f ).

Now note that for each i,f (x)hi(x)≤ hi(x)(ti + ε).

If x /∈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

(n

∑i=1

hi

).

Now note that |t0|+ ti + ε ≥ 0 and so from the definition of µ and Lemma 8.2.4, this is nolarger than

n

∑i=1

(|t0|+ ti + ε)µ(Vi)−|t0|µ(spt( f ))