26.1. THE FUNDAMENTAL THEOREM OF CALCULUS 937

Corollary 26.1.7 If f ∈ L1loc(Rn), then

limr→0

1m(B(x,r))

∫B(x,r)

f (y)dy = f (x) a.e. x. (26.1.3)

Proof: ∣∣∣∣ 1m(B(x,r))

∫B(x,r)

f (y)dy− f (x)∣∣∣∣

≤ 1m(B(x,r))

∫B(x,r)

| f (y)− f (x)|dy

and the last integral converges to 0 a.e. x.

Definition 26.1.8 For N the set of Theorem 26.1.5 or Corollary 26.1.6, NC is called theLebesgue set or the set of Lebesgue points.

The next corollary is a one dimensional version of what was just presented.

Corollary 26.1.9 Let f ∈ L1(R) and let

F(x) =∫ x

−∞

f (t)dt.

Then for a.e. x, F ′(x) = f (x).

Proof: For h > 0

1h

∫ x+h

x| f (y)− f (x)|dy≤ 2(

12h

)∫ x+h

x−h| f (y)− f (x)|dy

By Theorem 26.1.5, this converges to 0 a.e. Similarly

1h

∫ x

x−h| f (y)− f (x)|dy

converges to 0 a.e. x.∣∣∣∣F(x+h)−F(x)h

− f (x)∣∣∣∣≤ 1

h

∫ x+h

x| f (y)− f (x)|dy (26.1.4)

and ∣∣∣∣F(x)−F(x−h)h

− f (x)∣∣∣∣≤ 1

h

∫ x

x−h| f (y)− f (x)|dy. (26.1.5)

Now the expression on the right in 26.1.4 and 26.1.5 converges to zero for a.e. x. Therefore,by 26.1.4, for a.e. x the derivative from the right exists and equals f (x) while from 26.1.5the derivative from the left exists and equals f (x) a.e. It follows

limh→0

F(x+h)−F(x)h

= f (x) a.e. x

This proves the corollary.