Let
- $\lambda$ be the Lebesgue measure on the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R})$
- $f:\mathbb{R}\to\mathbb{R}$ be $\lambda$-integrable
What's the easiest way to show $$\frac d{dx}\int_{[0,x]}f\;d\lambda=f(x)\;\;\;\text{for }\lambda\text{-almost every }x\in\mathbb{R}\;?$$