$f$ integrable $\implies g(x) = \int_{-\infty}^x f$ is absolutely continuous

Question

Suppose that $f : \mathbb{R} \to \overline{\mathbb{R}}$ is an integrable function. Show that the function $g : \mathbb{R} \to \mathbb{R}$ defined by $g(x) = \int_{-\infty}^x f$ is absolutely continuous.

I was wondering if I could get a hint.

Answer 1

Hint: Let $m$ be the Lebesgue measure and define a measure $d\varphi = f dm$. The measure $\varphi$ is absolutely continuous with respect to the Lebesgue measure $m$. Apply this fact to the definition of absolute continuity of functions.