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 f\in L^{1}\left(\mathbb{R}^{d}\right), or more generally if f is locally integrable, then

    \[ \lim_{r\rightarrow0}\frac{1}{m\left(B\left(x,r\right)\right)}\int_{B\left(x,r\right)}f\left(y\right)dy=f\left(x\right), \]

for almost every x.

Proof. Let us denote by f_{r} the function

    \[ f_{r}\left(x\right)=\frac{1}{m\left(B\left(x,r\right)\right)}\int_{B\left(x,r\right)}f\left(y\right)dy,\quad r>0. \]

First, we claim if f\in L^{1}\left(\mathbb{R}^{d}\right), then \lim_{r\rightarrow0}\norm{f_{r}-f}1=0.
Note that for any r>0,

    \begin{align*} f_{r}\left(x\right) & =\frac{1}{m\left(B\left(x,r\right)\right)}\int_{B\left(x,r\right)}f\left(y\right)dy\\ & =\frac{1}{m\left(B\left(x,r\right)\right)}\int_{B\left(x,r\right)}f\left(-y\right)dy\\ & =\frac{1}{\omega_{d}r^{d}}\int_{B\left(0,r\right)}f\left(x-y\right)dy\\ & =\int_{\mathbb{R}^{d}}f\left(x-y\right)\frac{\chi_{B\left(0,r\right)}\left(y\right)}{\omega_{d}r^{d}}dy. \end{align*}

If we define \varphi\left(x\right)=\frac{\chi_{B\left(0,1\right)}\left(x\right)}{\omega_{d}}, then \varphi\ge0 and \int_{\mathbb{R}^{d}}\varphi\left(x\right)dx=1.
Define \varphi_{r}\left(x\right)=r^{-d}\varphi\left(x/r\right). Then \left\{ \varphi_{r}\right\} _{r>0} forms an approximation to the identity.

So

    \begin{align*} f_{r}\left(x\right) & =\int_{\mathbb{R}^{d}}f\left(x-y\right)\varphi_{r}\left(y\right)dy\\ & =\left(f*\varphi_{r}\right)\left(x\right). \end{align*}

This implies \lim_{r\rightarrow0}\norm{f_{r}-f}1=0. Therefore, f_{r_{k}}\rightarrow f, almost everywhere for a suitable sequence \left\{ r_{k}\right\} \rightarrow0. We are left to show that \lim_{r\rightarrow0}f_r\left(x\right) exists almost everywhere.

For this purpose, let us denote for each g\in L^{1}, and x\in\mathbb{R}^{d},

    \[ \omega\left(g\right)\left(x\right)=\left|\limsup_{r\rightarrow0}g_{r}\left(x\right)-\liminf_{r\rightarrow0}g_{r}\left(x\right)\right| \]

where g_{r} is defined like f_{r}. The above quantity represents the oscillation of the family \left\{ g_{r}\right\}, as r\rightarrow0.

Before to prove our theorem, we investigate the property of \omega\left(g\right)\left(x\right) first. If g is continuous with compact support, then g_{r}\rightarrow g uniformly since as we saw before,

    \[ g_{r}\left(x\right)=\left(f*\varphi_{r}\right)\left(x\right) \]

and \left\{ \varphi_{r}\right\} _{r>0} is a family of good kernels since it is an approximation to the identity. Since g is continuous with compact support, g_{r} converges uniformly to g.

If g\in L^{1}\left(\mathbb{R}^{d}\right), then by weak-type inequality, we have

    \[ m\left(\left\{ x:2M\left(g\right)\left(x\right)>\varepsilon\right\} \right)\le\frac{2A}{\varepsilon}\norm g1. \]

However, since g_{r}\left(x\right)\le M\left(g\right)\left(x\right)
for all r>0, we have

    \[ \omega\left(g\right)\left(x\right)\le2M\left(g\right)\left(x\right). \]

Since C_{0}^{\infty}\left(\mathbb{R}^{d}\right) is dense in L^{1}\left(\mathbb{R}^{d}\right), every f\in L^{1}\left(\mathbb{R}^{d}\right) can be written as f=h+g where h is continuous with compact support and g=f-h\in L^{1}\left(\mathbb{R}^{d}\right).

Note that \omega\left(f\right)\le\omega\left(h\right)+\omega\left(g\right) and \omega\left(h\right)=0 since h is continuous with compact support. So

    \[ m\left(\left\{ x:\omega\left(f\right)\left(x\right)>\varepsilon\right\} \right)\le\frac{2A}{\varepsilon}\norm g1. \]

Since the norm of g can be chosen to be arbitrarily small, we get \omega f=0 almost everywhere. So we are done.

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

  1. 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.

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