Step function approximation with Henstock–Kurzweil integral.

Question

In the following I am working with the Henstock–Kurzweil integral. I would like to prove the following:

Given a function $f : \mathbb{R} \rightarrow \mathbb{C}$ integrable on $[a..b]$, we have for any $e>0$ a tagged partition of $[a..b]$ and a step function $h$ (such that the value of $h$ across an interval in the partition is the value of $f$ at the tag of the interval) we have: $$ \int_a^b | f(x) - h(x) |\ dx < e $$

Answer 1

I think you cannot exchange absolute bar with integral since what you have to show is smaller than the condition written above.