lebesgue measure - Continuous function with derivative a.e. and Luzin N property is absolutely continuous

Let $f:I\rightarrow\mathbb{R}$ be a continuous function which derivative exists a.e. and $\int f'$ exists in I. Also, suppose $f$ satisfies the Luzin N property. Show that $f$ is absolutely continuous.

So I don't seem to find any way to tackle this problem. This seems to be covered everywhere when the derivative is defined everywhere, but no such case when it's only a.e.

What I know is that using bounded variation it's possible to get absolute continuity, however I can't find how to prove the bounded variation condition.

Is there a reference or a way to solve this problem?

Answer

See 2. here. The theorem that's being proven assumes bv, but that condition is only used to establish that $f$ has a derivative a.e. that is integrable, which are exactly the conditions you have.