# 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 LebesgueDifferentiationTheorem 2013-03-22 13:27:36 2013-03-22 13:27:36 Koro (127) Koro (127) 9 Koro (127) Theorem msc 28A15