192 CHAPTER 8. MEASURES AND MEASURABLE FUNCTIONS

Definition 8.7.2 If (X ,F ,µ) is a measure space, it is called σ finite if there areXn ∈F with ∪nXn = X and µ (Xn) < ∞. Note that by considering Yn = Xn \Xn−1,X0 ≡ /0we could assume X = ∪nYn where the Yn are disjoint.

Then there is a useful general result.

Theorem 8.7.3 Let (X ,d) be a metric space and suppose µ is σ finite and outerregular. Then µ is inner regular. If every closed set is the countable union of compact sets,then in the definition of inner regular, one can replace “closed” with “compact”.

Proof: Whenever µ (F) ,µ (G) < ∞ for G ⊇ F,µ (G\F) = µ (G)− µ (F) . I will usethis simple observation without comment in the following. To show the measure space isregular, the following picture might help or it might not. V is between the dotted lines.

V C∩K

V

U \FF

Let F be a bounded measurable set and let µ (U \F) < ε where U is open and letK ⊆ U, K closed and µ (U \K) < ε . I can get such a K because every open set is thecountable union of closed sets

U = ∪∞k=1

{x : dist

(x,UC)≥ 1

k

}≡ ∪∞

k=1Kk, ...Kk ⊆ Kk+1...

thus µ (U) < µ (Kk) for all k large enough since µ (U) = limk→∞ µ (Kk) by Lemma 8.2.4.Then let V be open and µ(V \ (U \F)) < ε where V ⊇U \F = U ∩FC so VC ⊆UC ∪F .This is possible because all sets are in F . Then VC ∩K ⊆

(UC ∪F

)∩K = F ∩K ⊆ F .

Now VC ∩K is compact and

µ(F \(K∩VC)) = µ

(F ∩

(KC ∪V

))= µ (F ∩V )+µ

(F ∩KC)

≤ µ (F ∩V )+µ (U \K)< µ (F ∩V )+ ε

However, ε > µ(V \ (U \F)) = µ

(V ∩

(U ∩FC

)C)= µ

(V ∩

(UC ∪F

))≥ µ (V ∩F) and

so µ(F \(K∩VC

))≤ 2ε . That µ (F) = sup{µ (K) : K ⊆ F} follows from observing that

µ (F) = limn→∞ µ (F ∩B(0,n)) and then applying what was just shown to a suitable F ∩B(0,n). As to the last claim, it follows from observing that for K a closed set, there is anincreasing sequence of compact sets {Kn} whose union is K and then using Lemma 8.2.4.■

The following is a nice formulation of the above and also gives a useful claim aboutuniqueness.

Theorem 8.7.4 Suppose (X ,F ,µ) ,F ⊇B (X) is a measure space for X a metricspace and µ is σ finite, X = ∪nXn with µ (Xn)< ∞ and the Xn disjoint Borel sets. Supposealso that µ is outer regular. Then for each E ∈ F , there exists F,G an Fσ and Gδ set