Lebesgue differentiation theorem

Let $f$ be a locally integrable function on $\mathbb{R}^{n}$ with Lebesgue measure $m$ , i.e. $f\in L^{1}_{\textnormal{loc}}(\mathbb{R}^{n})$ . Lebesgue’s differentiation theorem basically says that for almost every $x$ , the averages

$\frac{1}{m(Q)}\int_{Q}|f(y)-f(x)|dy$

converge to $0$ when $Q$ is a cube containing $x$ and $m(Q)\rightarrow 0$ .

Formally, this means that there is a set $N\subset\mathbb{R}^{n}$ with $\mu(N)=0$ , such that for every $x\notin N$ and $\varepsilon>0$ , there exists $\delta>0$ such that, for each cube $Q$ with $x\in Q$ and $m(Q)<\delta$ , we have

$\frac{1}{m(Q)}\int_{Q}|f(y)-f(x)|dy<\varepsilon.$

For $n=1$ , this can be restated as an analogue of the fundamental theorem of calculus for Lebesgue integrals. Given a $x_{0}\in\mathbb{R}$ ,

$\frac{d}{dx}\int_{x_{0}}^{x}f(t)dt=f(x)$

for almost every $x$ .

Title	Lebesgue differentiation theorem
Canonical name	LebesgueDifferentiationTheorem
Date of creation	2013-03-22 13:27:36
Last modified on	2013-03-22 13:27:36
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	9
Author	Koro (127)
Entry type	Theorem
Classification	msc 28A15