Show that $f$ is measurable on $(a,b)$

Question

Let $f,F : (a,b)\to \mathbb{R}$ such that $F$ is differentiable on $(a,b)$ and $F'(x) =f(x)$ for each $x \in (a,b)$. Show that $f$ is measurable on $(a,b)$. Hint : Show first that $\forall x \in (a,b) : f(x) = \lim\limits_{n \to \infty} n[F(x+\frac{1}{n}) - F(x)]$.

I have already proved the hint ... How to proceed please?

Answer 1

Note that each function $x\mapsto n(F(x + 1/n) - F(x))$ is measurable and that the limit of a sequence of measurable functions is measurable.