A classic exercise is to show that integration is absolutely continuous. That is, for $f\in L^1(\mathbb{R}^n,\mu)$ and for all $\varepsilon>0$ there exists $\delta>0$ such that whenever the measure of $A\subset\mathbb{R}^n$ is less that $\delta$, I.e. $\mu(A)<\delta$, then one has $$\int_A |f|d\mu<\varepsilon$$
What I’m particularly interested in is if one can say anything about how fast the integral goes to zero with measure of the set. That is, can one say anything about the rate of convergence of the limit, $\lim_{\mu(A)\to0}\int_A|f|d\mu$?
If not, does anyone know of any conditions under which we can say something of the type?