Let $f$ be a locally integrable function on $\mathbb{R}^n$ with Lebesgue measure $m$ i.e. $f\in L^1_{loc}(\mathbb{R}^n)$ 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 $x$