Let $F:\mathbb{R}\rightarrow [0, 1]$ be a increasing function in $\mathbb{R}$ hat is strictly increasing in $[0, \infty[$ such that $F(0)=0$. Suppose that $F$ is absolutely continuous in $\mathbb{R}$ i.e. $F(x)=\int_{0}^{x}{f(t)dt}$.
How can I prove that $F*f$ is absolutely continuous?
What assumptions do I have to do with $f$ in order to prove that?
And lastly, How to prove that the series $\sum_{n=1}^{\infty}{F*f^{*(n)}(x)}$ is convergent?
I figure that for the first part, I have to express $F*f$ like an indefinite integral, and maybe Fubini can help.