8.4. LEBESGUE MEASURE 195

Proof: From Corollary 8.2.8, if E ∈F , there exists {Kn} be an increasing sequence ofcompact sets such that

µ (E) = limn→∞

µ (Kn) .

Then if F ≡∪nKn, it follows that F is an Fσ set and µ (F) = limn→∞ µ (Kn) = µ (E). Thus,in particular F is a Borel set. Actually, one can say a little more. Note that, by assumption,µ (B) < ∞ for any ball since µ (K) < ∞ for any compact set and B is compact. Let x begiven in X and let Bn ≡ B(x,n). Let En ≡ E ∩ Bn so µ (En) < ∞. From what was justshown, there exists an Fσ set Fn ⊆ En such that µ (En) = µ (Fn) . Since the measures arefinite, µ (En \Fn) = 0. Then letting F ≡ ∪∞

n=1Fn, it follows that this new F is an Fσ set and

µ (E \F) = µ (∪nEn \∪nFn)≤ µ (∪n (En \Fn))

≤ ∑n

µ (En \Fn) = 0.

Let E,En be as above. Using outer regularity, there is an open set Vn containing En suchthat µ (Vn \En)< ε2−n. Let Wε ≡∪nVn. Thus µ (Wε \E)≤ µ

(∪∞

k=1 (Vk \Ek))≤ ε and Wε

is open andµ (Wε)< ε +µ (E) , µ (Wε \E)< ε

It follows there exists a decreasing sequence of open sets Wn each containing E such thatµ (Wn)< 2−n+µ (E) , and µ (Wn \E)< 2−n. Let G≡∩nWn. Then G is a Gδ set containingE and for each n,

µ (G\E)≤ µ (Wn \E)< 2−n

and so µ (G\E) = 0 which implies µ (G) = µ (E) . Now µ (G) = µ (E) = µ (F) . Also, theFσ set F from the first part with µ (E \F) = 0,

µ (G\F) = µ (G\E)+µ (E \F) = 0

This proves the first part.For the remaining part, it suffices to consider only f (x)≥ 0 because you can reduce to

positive and negative parts of real and imaginary parts of f . By Theorem 6.1.10, there is anincreasing sequence of simple functions sk such that for all x,sk (x) ↑ f (x). Now for sk (x) =∑

mki=1 ak

i XEki,(ak

i > 0) replace each Eki with Fk

i an Fσ set with µ(Ek

i \Fki)= 0,Fk

i ⊆ Eki . Let

Nk ≡ ∪mki=1

(Ek

i \Fki)

and let N̂ ≡ ∪kNk a set of measure zero. Thus there exists a Borel setN ⊇ N̂ which also has measure zero, this by the first part. In fact, we can take N to be aGδ set. Let ŝk (x) = sk (x) off N and let ŝ(x) = 0 on N. Thus ŝk (x) is an increasing functionwhich converges to f (x) off the set of measure zero N and converges to 0 on N. Each ŝkis Borel measurable and so letting g be the pointwise limit, it follows from Corollary 6.1.4that g is Borel measurable and 0≤ g≤ f . ■

The above approximation result applies to any of the measures from Theorem 8.2.1.Next is a specialization to Lebesgue measure on Rp.

8.4 Lebesgue MeasureNow we define Lebesgue measure in terms of a functional from beginning calculus.

Definition 8.4.1 Lebesgue measure, called mp is obtained from using the aboveRiesz representation theorem for positive linear functionals on the functional

L f ≡∫

∞

−∞

· · ·∫

∞

−∞

f (x1, · · · ,xp)dxpdxp−1 · · ·dx1