Gronwall Lemma, understanding Proof

Question

Let $a \in L^{\infty}(t_0,T)$ and $\lambda \in L^1(t_0,T)$ where $[t_0,T]\subset \mathbb{R}$.

Consider $f(t):=e^{\int_{t_0}^t \lambda(s) ds}\, \int_{t_0}^t \lambda(s) a(s) ds, \, t \in [t_0,T]$.

Why is $f(t)$ absolutely continuous?

Answer 1

Hints: On a compact interval the following facts are true:

1) Indefinite integral of an integrable function is absolutely continuous.

2) If $f$ is absolutely continuous and bounded then so is $e^{f}$. [For this use MVT and definition of absolutely continuity].

3) Product of two absolutely continuous functions is absolutely continuous.