One part of Theorem 14 (Chapter 5) of the book Real Analysis (3rd edition) by Royden, says that:
if a function $F$ is an indefinite integral, then it is absolutely continuous.
The proof says that this statement is obvious using the following Proposition:
Let $f$ be a non-negative function which is integrable over a set $E$. Then given $\varepsilon \gt 0$, there is a $\delta \gt 0$ such that for every set $A\subset E$ with the $m(A)\lt \delta$, we have $$ \int_A f \lt \varepsilon.$$
I am failing to see how to use the above proposition in the proof and thus will need some help.