10.2. FUNDAMENTAL THEOREM OF CALCULUS 239

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 8.4.2.mp (E)≡ inf

{mp (F) : F is measurable and F ⊇ E

}.

Theorem 10.2.1 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 4.5.3.

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. (10.1)

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 10.1,

mp ([M f > ε])≤∑i

mp(B̂i)≤ 5p

∑i

mp (Bi)≤5p

ε∑

i

∫Bi

| f |dmp ≤ 5pε−1 ∥ f∥1

This proves claim 1.Claim 2: If g ∈Cc (Rp), then

limr→0

1mp (B(x,r))

∫B(x,r)

|g(y)−g(x)|dmp (y) = 0

Proof: Since g is continuous at x, whenever r is small enough,

1mp (B(x,r))

∫B(x,r)

|g(y)−g(x)|dmp (y)≤1

mp (B(x,r))

∫B(x,r)

ε dmp (y) = ε.

This proves the claim.Now let g ∈Cc (Rp). Then from the above observations about continuous functions in

Claim 2,

mp

([x : limsup

r→0

1mp (B(x,r))

∫B(x,r)

| f (y)− f (x)|dmp (y)> ε

])(10.2)