Absolute continuity of the Lebesgue integral
= Absolute continuity of the Lebesgue integral
If $f\in L^1$, then for every $\varepsilon>0$ there is $\delta>0$ such that $\int_E|f|<\varepsilon$ whenever the measure of $E$ is below $\delta$. In particular, integrals over a sequence of shrinking measurable sets tend to zero.