Lebesgue Differentiation Theorem

The Lebesgue Differentiation Theorem states:

If is a summable function then for almost every the following limit converges to :

Where is the lebesgue measure and is an open ball of radius .

How does it follow that this Theorem is satisfied if is only assumed to be continuous?

Let me know if anything is unclear. Thanks. (Headbang)

Re: Lebesgue Differentiation Theorem

I believe it is required that be Lebesgue integrable.

See the proof here.

Re: Lebesgue Differentiation Theorem

Quote:

Originally Posted by

**Kramer** The Lebesgue Differentiation Theorem states:

If

is a summable function then for almost every

the following limit converges to

:

Where

is the lebesgue measure and

is an open ball of radius

.

How does it follow that this Theorem is satisfied if

is only assumed to be continuous?

When f is (additionaly) continuous, then it is uniformly continuous over the closed balls . Choose such a ball of radius . Then, due to uniform continuity, for every there is such that

This implies

from where the result follows.