Almost everywhere differentiability of $\int_{0}^{x}a(t)\,dt.$

Question

Let $a(x):\mathbb{R}_+\to\mathbb{R}$ be locally integrable function in Lebesgue sense and $$A(x)=\int_{0}^{x}a(t)\,dt.$$

Let $A(x)$ be differentiable almost everywhere. Is it always true that $A^\prime(x)=a(x)$.

Answer 1

If $a$ is locally integrable then $A'(x)=\lim_{h \to 0} \frac 1 h \int_ x^{x+h} a(t)dt=a(x)$ whenever $x$ is a Lebesgue point of $a$. Almost all points are Lebesgue points so $A'(x)=a(x)$ almost everywhere.

For Lebesgue points see https://en.wikipedia.org/wiki/Lebesgue_point