192 CHAPTER 8. POSITIVE LINEAR FUNCTIONALS

≤n

∑i=1

(|t0|+ ti + ε)(µ(Ei)+ ε/n)−|t0|µ(spt( f ))

≤ |t0|

µ(spt( f ))︷ ︸︸ ︷n

∑i=1

µ(Ei)+ε

nn |t0|+∑

itiµ (Ei)

+∑i

tiε

n+∑

iεµ (Ei)+

ε2

n−|t0|µ(spt( f ))

≤ ε |t0|+ ε (|t0|+ |b|)+ εµ(spt( f ))+ ε2 +∑

itiµ (Ei)

≤ ε |t0|+ ε (|t0|+ |b|)+2εµ(spt( f ))+ ε2 +

n

∑i=1

ti−1µ(Ei)

≤ ε (2 |t0|+ |b|+2µ(spt( f ))+ ε)+∫

f dµ

Since ε > 0 is arbitrary, L f ≤∫

f dµ for all f ∈Cc(X), f real. Hence equality holds because

−L( f ) = L(− f )≤∫

(− f )dµ =−∫

f dµ

so L( f ) ≥∫

f dµ . Thus L f =∫

f dµ for all f ∈Cc(X). Just apply the result for real func-tions to the real and imaginary parts of f . ■

In fact the two outer measures are equal on all sets. Thus the measurable sets are exactlythe same and so they have the same σ algebra of measurable sets and are equal on this σ

algebra. Of course, if you were willing to consider σ algebras for which the measuresare not complete, then you might have different σ algebras, but note that if you define themeasurable sets in terms of Caratheodory as done above, the σ algebras are also unique.

As a special case of the above,

Corollary 8.2.8 Let µ be a Borel measure. Also let µ (B) < ∞ for every ball B con-tained in a metric space X which has closed balls compact. Then µ must be regular.

Proof: This follows right away from using the Riesz representation theorem above onthe functional

L f ≡∫

f dµ

for all f ∈Cc (X). ■Here is another interesting result.

Corollary 8.2.9 Let X be a random variable with values in Rp. Then λ X is an innerand outer regular measure defined on B (Rp).

This is obvious when you recall the definition of the distribution measure for the randomvector X. λ X (E) ≡ P(ω : X(ω) ∈ E) where this means the probability that X is in E aBorel set of Rp. It is a finite Borel measure and so it is regular.

There is an interesting application of regularity to approximation of a measurable func-tion with one that is continuous.