Absolute continuity of the integral of an $L^1$ function

For $f\in L^1$ and $\varepsilon>0$, $\int_E\abs{f}<\varepsilon$ for every set $E$ of sufficiently small measure.Suggest a slogan

Let $m$ be Lebesgue measure on $\RR^n$ and let $f\in L^1(\RR^n)$. For every $\varepsilon>0$ there exists $\delta>0$ such that every Lebesgue measurable set $E\subseteq\RR^n$ with $m(E)<\delta$ satisfies \[ \abs{\int_E f}\le\int_E \abs{f} < \varepsilon . \]