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.
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
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.