9.5. FUNDAMENTAL THEOREM OF CALCULUS 231

This shows that if f is zero off some ball, then it can be approximated with a continuousfunction which is zero off a compact set.

Now consider the general case. Then let fn = f XB(0,n). Then by dominated conver-

gence theorem, for n large enough, (∫| f − fn|p dµ)1/p < ε

2 and now from what was justshown, there exists h continuous, zero off some compact set, such that (

∫| fn−h|p dµ)1/p <

ε

2 . Thus from the triangle inequality,(∫| f −h|p dµ

)1/p

<

(∫| f − fn|p dµ

)1/p

+

(∫| fn−h|p dµ

)1/p

< ε.■

9.5 Fundamental Theorem of CalculusIn this section the Vitali covering theorem, Proposition 8.6.3 will be used to give a gener-alization of the fundamental theorem of calculus. Let f be in L1 (Rp) where the measure isLebesgue measure as discussed above.

Let M f : Rp→ [0,∞] by

M f (x)≡ supr≤1

1mp (B(x,r))

∫B(x,r)

| f |dmp if x /∈ Z.

We denote as ∥ f∥1 the integral∫

Ω| f |dmp.

The special points described in the following theorem are called Lebesgue points. Alsomp will denote the outer measure determined by Lebesgue measure. See Proposition 6.4.2.mp (E)≡ inf

{mp (F) : F is measurable and F ⊇ E

}.

Theorem 9.5.1 Let mp be p dimensional Lebesgue measure measure and let f ∈L1 (Rp,mp).(

∫Ω| f |dmp < ∞). Then for mp a.e.x,

limr→0

1mp (B(x,r))

∫B(x,r)

| f (y)− f (x)|dmp (y) = 0

Proof: First consider the following claim which is called a weak type estimate.Claim 1: The following inequality holds for Np the constant of the Vitali covering

theorem, Proposition 8.6.3.

mp ([M f > ε])≤ 5pε−1 ∥ f∥1

Proof: For each x ∈ [M f > ε] there exists a ball Bx = B(x,rx) with 0 < rx ≤ 1 and

mp (Bx)−1∫

B(x,rx)| f |dmp > ε. (9.7)

Let F be this collection of balls. By the Vitali covering theorem, there is a collection ofdisjoint balls G such that if each ball in G is enlarged making the center the same but theradius 5 times as large, then the corresponding collection of enlarged balls covers [M f > ε] .By separability, G is countable, say {Bi}∞

i=1 and the enlarged balls will be denoted as B̂i.Then from 9.7,

mp ([M f > ε])≤∑i

mp(B̂i)≤ 5p

∑i

mp (Bi)≤5p

ε∑

i

∫Bi

| f |dmp ≤ 5pε−1 ∥ f∥1