# Lebesgue differentiation theorem with approximation to the identity

By | June 2, 2015

Lebesgue differentiation theorem은 Riemann integration에서 fundamental theorem of calculus를 measure theory적인 언어로 확장한 정리다. 이를 증명하는 방법이 여러가지가 있으나, 이 절에서는 Stein의 Singular integral and differential properties of functions에 있는 방법을 소개하고자 한다. 이 증명은 약간 good kernel스러운 방법을 쓰고자 한다.

Lebesgue Differentiation Theorem. If , or more generally if is locally integrable, then for almost every .

Proof. Let us denote by the function First, we claim if , then .
Note that for any , If we define , then and .
Define . Then forms an approximation to the identity.

So This implies . Therefore, , almost everywhere for a suitable sequence . We are left to show that exists almost everywhere.

For this purpose, let us denote for each , and , where is defined like . The above quantity represents the oscillation of the family , as .

Before to prove our theorem, we investigate the property of first. If is continuous with compact support, then uniformly since as we saw before, and is a family of good kernels since it is an approximation to the identity. Since is continuous with compact support, converges uniformly to .

If , then by weak-type inequality, we have However, since for all , we have Since is dense in , every can be written as where is continuous with compact support and .

Note that and since is continuous with compact support. So Since the norm of can be chosen to be arbitrarily small, we get almost everywhere. So we are done.

## 2 thoughts on “Lebesgue differentiation theorem with approximation to the identity”

1. 성기훈

ㅋㅋㄱㅋ이거군요….흐….역시

2. 지나가는유저

latex 관련해서 검색하다가 저랑 같은 닉네임을 쓰는 분이 계셔서 방문했습니다.
제가 10살때.. 그러니까 17년 전에 아버지가 제게 첨으로 지어준 인터넷 아이디가 skykite였습니다.
신기하네요. 저희 아버지도 수학자고 주 연구분야가 조화해석학이거든요.

건승을 빕니다.

This site uses Akismet to reduce spam. Learn how your comment data is processed.