280 CHAPTER 12. THE CONSTRUCTION OF MEASURES

There is an interesting situation in which regularity is obtained automatically. To saveon words, let B (E) denote the σ algebra of Borel sets in E, a closed subset of Rn. It is avery interesting fact that every finite measure on B (E) must be regular.

Lemma 12.1.11 Let µ be a finite measure defined on B (E) where E is a closed subset ofRn. Then for every F ∈B (E) ,

µ (F) = sup{µ (K) : K ⊆ F, K is closed }

µ (F) = inf{µ (V ) : V ⊇ F, V is open}

Proof: For convenience, I will call a measure which satisfies the above two conditions“almost regular”. It would be regular if closed were replaced with compact. First noteevery open set is the countable union of compact sets and every closed set is the countableintersection of open sets. Here is why. Let V be an open set and let

Kk ≡{

x ∈V : dist(x,VC)≥ 1/k

}.

Then clearly the union of the Kk equals V and each is closed because x→ dist(x,S) isalways a continuous function whenever S is any nonempty set. Next, for K closed let

Vk ≡ {x ∈ E : dist(x,K)< 1/k} .

Clearly the intersection of the Vk equals K. Therefore, letting V denote an open set and K aclosed set,

µ (V ) = sup{µ (K) : K ⊆V and K is closed}µ (K) = inf{µ (V ) : V ⊇ K and V is open} .

Also since V is open and K is closed,

µ (V ) = inf{µ (U) : U ⊇V and V is open}µ (K) = sup{µ (L) : L⊆ K and L is closed}

In words, µ is almost regular on open and closed sets. Let

F ≡{F ∈B (E) such that µ is almost regular on F} .

Then F contains the open sets. I want to show F is a σ algebra and then it will followF = B (E).

First I will show F is closed with respect to complements. Let F ∈F . Then since µ isfinite and F is inner regular, there exists K⊆F such that µ (F \K)< ε. But KC \FC =F \Kand so µ

(KC \FC

)< ε showing that FC is outer regular. I have just approximated the

measure of FC with the measure of KC, an open set containing FC. A similar argumentworks to show FC is inner regular. You start with V ⊇ F such that µ (V \F) < ε , noteFC \VC = V \ F, and then conclude µ

(FC \VC

)< ε, thus approximating FC with the

closed subset, VC.