11.6. MEASURES AND REGULARITY 259

Definition 11.6.2 A measure µ defined on B (E) will be called inner regular if for allF ∈B (E) ,

µ (F) = sup{µ (K) : K ⊆ F and K is closed} (11.6.26)

A measure, µ defined on B (E) will be called outer regular if for all F ∈B (E) ,

µ (F) = inf{µ (V ) : V ⊇ F and V is open} (11.6.27)

When a measure is both inner and outer regular, it is called regular. Actually, it is moreuseful and likely more standard to refer to µ being inner regular as

µ (F) = sup{µ (K) : K ⊆ F and K is compact} (11.6.28)

Thus the word “closed” is replaced with “compact”. A complete measure defined on a σ

algebra F which includes the Borel sets which is finite on compact sets and also satisfies11.6.27 and 11.6.28 for each F ∈F is called a Radon measure.

For finite measures, defined on the Borel sets of a metric space, the first definition ofregularity is automatic. These are always outer and inner regular provided inner regularityrefers to closed sets.

Lemma 11.6.3 Let µ be a finite measure defined on B (X) where X is a metric space.Then µ is regular.

Proof: First note every open set is the countable union of closed sets and every closedset is the countable intersection 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. Thus

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

If U is open and contains V, then µ (U)≥ µ (V ) and so

µ (V )≤ inf{µ (U) : U ⊇V, U open} ≤ µ (V ) since V ⊆V.

Thus µ is inner and outer regular on open sets. In what follows, K will be closed and Vwill be open.

Let K be the open sets. This is a π system. Let

G ≡ {E ∈B (X) : µ is inner and outer regular on E} so G ⊇K .

For E ∈ G , let V ⊇ E ⊇ K such that µ (V \K) = µ (V \E)+µ (E \K)< ε . Thus KC ⊇ EC

and so µ(KC \EC

)= µ (E \K)< ε. Thus µ is outer regular on EC because

µ(KC)= µ

(EC)+µ

(KC \EC)< µ

(EC)+ ε, KC ⊇ EC