Lebesgue differentiation theorem

Let $f$ be a locally integrable function on ${ℝ}^n$ with Lebesgue measure $m$, i.e. $f\in L_{\text{loc}}^1({ℝ}^n)$. 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 $m(Q)\to 0$.

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

For $n=1$, this can be restated as an analogue of the fundamental theorem of calculus for Lebesgue integrals. Given a $x_0\in ℝ$,

$$\frac{d}{dx}{\int }_{x_0}^{x}f(t)𝑑t=f(x)$$

for almost every $x$.